Themself was used in the past, and there is no law or authority that prohibits anyone from using it today. I have used it in personal correspondence, conscious of its rebellious and contradictory nature; however, I have to confess many of my correspondents are in the field of language teaching, and they tend to be more open-minded.
Although the singular themself is gaining currency, it would be an arduous challenge for anyone to produce a recent government bill, act, tax form, or any official English document that contains the actual reflexive pronoun. And if they could produce a formal document, it would be akin to seeing an exotic and engendered butterfly in the wild.
It's simply not done; not today, not in a formal context simply because it looks “wrong”. Themself looks dialectal, a word that an uneducated native speaker person might use. While the singular they, their and them are extremely common in speech—and increasingly so in writing as it avoids having to write the cumbersome he or she; his or her; him or her—yet many English native speakers consider themself not a “proper word”, and whenever instances of ourself and themself appear in writing, these words stick out like a sore thumb.
Those in favour of “themself”
Pam Peters in ‘The Cambridge Guide to English Usage’ advocates:
The singular reference in ‘themself’ obviously serves a purpose, especially after an indefinite noun or pronoun. If we allow the use of ‘they’/’them’/’their’ for referring to the singular, ‘themself’ seems more consistent than ‘themselves‘. We make use of ”yourself‘ alongside ‘yourselves’ in just the same way. ‘Themself’ has the additional advantage of being gender-free, and thus preferable to both ‘himself’ and ‘himself/herself‘. It’s time to reinstate it to the set of reflexive pronouns!
Those against …
From an article in Language Log, March 08, 2007, two American English authorities condemn the use of themself
- As MWDEU (Merriam-Webster Dictionary of English Usage) 1989 puts it (p. 898):
This use of themself is similar to the use of they, their, and them in reference to singular terms... Such use of they, their, and them is old and well established, but this use is not.
- Wilson's Columbia Guide (1993) is stern on the matter (p. 435):
Theirselves and themself for themselves are limited to Vulgar English or imitations of it; both are shibboleths.
Themself can also occur as an unfortunate result of trying to avoid using a gender-explicit reflexive pronoun by using a blend of the plural them with the singular self. The choices are themselves or himself or herself or both the last two...
An Ngram showing themself tells us that it existed and exists. An Ngram that compares themself and themselves reflects its usage more accurately.
Him(self) or herself
An Ngram that compares themself (blue line); himself or herself (red line); him or herself (green) and herself and himself (yellow) tells us that the majority of writers (and editors) feel more comfortable using a longer equivalent than the succinct themself.
On Google Books, the politically-correct expression, "herself or himself", produces around 1,480 results. Here are some examples:
The differentiation between self and not-self certainly seems related to the growth of the object concept, during which the child learns to see herself or himself as an object in space and time, separate from the mother.
Research Manual in Child Development 2003
1963, Standard Civil Code of the State of California
the case may be, for the permanent support and maintenance of  herself or himself, and may include therein at her or his discretion an action for support, maintenance and education of the children of said marriage during their minority.
and as recently as 2009, Code of Federal Regulations
(a) An ALJ [Administrative Law Judge] may disqualify herself or himself at any time. (b) Until the filing of the ALJ's decision. either party may move that the ALJ disqualify herself or himself for personal bias or other valid cause. The party shall file with the ALJ, promptly ..
Whereas himself or herself gets 8,190 hits
George Herbert Mead and Human Conduct, 2004
He sees it, in the first instance, as being merely the object that the individual is to himself or herself. Obviously, human beings can, and do, think of themselves as being a given kind of object. The human being may see himself or herself as male or female, young or old, rich or poor, married or unmarried …
Interestingly, the authors use the impersonal pronouns it and itself when referring to babies and small infants on page 58.
The human infant or very young child is not an object to itself. While in the eyes of others it acts as a baby, it doesn't recognize itself as a baby. It doesn't see itself as someone who is helpless, gets sick, cries a lot, spends a lot of time sleeping, ...
In a formal or technical register, himself or herself, will usually be preferred. And it seems highly unlikely that it will change in the near future.
Criminal Law, 2010, page 357
Section 2 Any person who
(a) Purposely engages in a course of conduct directed at a specific person that would cause a reasonable person to fear bodily injury to himself or herself or a member of his or her immediate family or to fear the death of himself or herself or a member of his or her immediate family; and
(b) Has knowledge or should have knowledge that the specific person will be placed in reasonable fear of bodily injury to himself or herself or a member of his or her immediate family or induce fear in the specific individual of the death of himself or herself or a member of his or her immediate family; is guilty of stalking.
And those sitting on the fence
In 2013, Catherine Soanes, guest blogger on OxfordWords blog, and one of the editors of the OED 2nd edition 2005, argued:
Given that it’s now largely acceptable to use they, them, or their instead of the more long-winded ‘he or she’, ‘him or her’, or ‘his or her’ (especially in conjunction with indefinite pronouns such as anyone or somebody) it might be argued that, logically, it should also be OK to use themself, it being viewed as the corresponding singular form of themselves. However, this isn’t yet the case, so beware of themself for now! The correct versions of the opening examples in this section should be:
- It’s not an expensive way for somebody to make themselves feel good.
- Anyone would find themselves thinking similar thoughts.
Of course, if you dislike the use of gender-neutral third-person plural pronouns for singular subjects, or you’re working to a style guide that prohibits them, you should reword the sentences so as to incorporate gender-specific third-person singular pronouns instead:
- It’s not an expensive way for somebody to make himself or herself feel good.
- Anyone would find himself or herself thinking similar thoughts.
[…] To sum up, the wheel has not yet come full circle and ‘themself’ remains a standard English outcast. . . for now.
If you dislike using “themself”, what can you do?
The OP's example:
If someone wants to watch TV tonight, they'll have to do the dishes
all by themselves
Sound perfectly acceptable to my ears. In speech and in an informal context, it is perfectly fine. For anyone who dislikes this solution I would suggest the following:
If I am speaking to more than one person
i) For those who want to watch TV tonight, they'll have to do the dishes all by themselves
or to any individual, male or female
ii) If someone wants to watch TV tonight, he or she will have to do the dishes
all by themselves
or you could try this "clunkier" version
iii) If someone wants to watch TV tonight, he or she will have to do the dishes all by him or herself
If I had to use this particular construction, for efficiency's sake, I'd choose him or herself, which is well-documented and represented by the green line in the third Ngram chart.