Is logicalising a valid word, if the meaning was to make something supposedly 'illogical', logical?
"Logicalise" already exist as a word, and it already means something other than the definition you listed. It means "to reason, to think, to use or apply logic", without the connotation of trying to make something illogical into something logical. For example, see here:
The word you might be looking for is "rationalise" or its various synonyms.
rationalise: attempt to explain or justify [something] with logical, plausible reasons, even if these are not true or appropriate. (from ODO)
'Logical' is an adjective which you have then verbalised. I would say 'logicate' is the correct verb from the noun, 'logic'.
And, since it is available in English, it is quite valid. I know of several books that have used 'logicate' within the past five years.
'Logicate' is similar to calculate, but 'calculate' is a matter of arithmetical logic whereas 'logicate' is the process which uses pure logic, rather than numbers.