Or maybe, the person's finished painting the room? Who knows! Finished doing/done with doing what in the children's study? And which children have their own study?
Sorry, it's just clicked from reading 2010's post that you probably don't mean a room: you mean studies - possibly, homework:
'I've almost finished helping the children with their studies.'
'I'm almost done with helping the children with their studies.'
Am I right?
In which case - well, in any case - the meaning is the same. I think British people would be more likely to say 'I've finished -ing'. To me, 'I'm done with -ing' is a needless colloquialism, although having said that, a simple 'I'm done!' to mean 'I'm/I've finished!' is quite common.
Also, I'm pretty sure 'I'm done ...' is used more in AmEng, but is that more usually without the 'with', so:
'I'm almost done helping the children with their studies.'?
That said, I'm pretty sure Americans would also be more likely to say 'I've finished' - wouldn't they?