(a) She gets up latest in her family.
(b) She gets up the latest in her family.
For me the "in her family" makes NEITHER of these acceptable - just as FumbleFingers commented I'd use last, preferring the form with "the family" to "her family" and "to get up" deferred.
- She's [always] the last in the family to get up
- She's [always] the last to get up in her family.
The reason is that this would normally be said as a complaint by a member of the same family (so it is not her family but our family, which is so close it is the only one that counts).
Then "in the family" would mostly be implied if you were at home saying this, but would be added if you were gossipping outside the home, and then "in her family" would be if you were repeating the gossip (only people in the household would know this otherwise).
Then we have "the last in the/her family" as the complement to "She is" - this is a single noun phrase. The versions with "in her family" last are possible but sound like an afterthought:
She's the last to get up ... in her family! But when she's staying with us, she's the first!
So now that we've disposed of "in her family" and the awkwardness that that lends, let's consider:
(a) She gets up latest.
(b) She gets up the latest.
The first one is less acceptable because this adverbial version of the superlative has been reduced to "last" when used in a relative sense rather than relative to a deadline (she is not necessarily late for anything). Normally before a superlative "the" will be required, and in the case of a singular referent (she - the latest riser) it has the import of "the one" - the unique individual singled out by the superlative, or in the case of the plural the unique group identified. The universe which limits the application of the superlative can be given by "in her family" or "amongst her peers" - and I would tend to put it first rather than last.
Conversely, in the case of a comparative, there is no such implication of uniqueness, but the comparison normally must be spelled out with than. Of course if it is everyone else in the universe specified with the superlative, there is no semantic difference, only subtle pragmatic differences that determine which of these variants has the appropriate focus of the context:
- In terms of the students in the dorm, she gets up the latest.
- She is always the latest to breakfast/the last up of the students in her dorm.
- She gets up later than the other students in her dorm.
Re the absolute/relative distinction between last and latest, consider
- She is always the last to class - but she is still always on time.
- She is always the latest to class - and there are quite a few who are late.
In your original two sentences, the adjective plays an adverbial role in the first, and a substantive (noun-like) role in the second, but then the whole noun phrase can carry an adverbial role (telling you something about how/when she gets up).
An analogous pair with an unambiguous noun forming an adverbial phrase expands out to:
She comes Tuesday; he comes Wednesday.
She comes Tuesdays; he comes Wednesdays.
She comes the Tuesday; he comes the Wednesday.
She comes the Tuesdays; he comes the Wednesdays.
The definite article would be used where there has already been discussion about when the various people come - or in the case of your sentences about when various people get up.
It is used to select a particular day (out of several) or a particular degree of lateness (out of several). You really need the full context to make either of your sentences sound natural - at the moment they are both awkward.