This is a pretty difficult puzzle! My solution here has a lot of 'usual' deductions left unmentioned.
So first of all, all the 1s can be closed off:
Now, there's one cell that must be focused on:
The top left cell cannot be a 1, so it must extend rightwards. It then cannot be a 2, so the 2 near the top must go down. The same logic can be applied again to get this far:
the 2 in the top middle. Can it extend downwards? If it does, then R3C5 must go left to make a size-3 flat region that touches the 3 clue. That's a problem -- so the 2 must go upwards. This forces the top left region to be a 5.
Now, an important question appears:
What's going on in the center? There can't be any 1s in there. There can't be any 2s either. And we can't put a size-3 region in there: so it must be filled by a 4!
Now, we can finish the upper right corner, and start on the lower right:
There must be a single isolated 1 in the upper right, and one of the two potential places is adjacent to an already-existing 1.
Meanwhile, R5C8 must be a 4, completing a 4 region. This lets us continue down the right-hand side.
Now, I make an assumption for sake of contradiction. (This part of the puzzle seems like it should have a 'cleaner' path, but I couldn't find one.)
Assume that the 3 clue near the center does not go down. It must then go left, and then either left or down: either way, the remaining empty cells in column 3 cannot be satisfied. (If left, R6C3 cannot be a 3 or a 1; if down, R8C3 cannot be a 1, 2, or 3, but it only has 3 cells of space to expand into.)
This allows us to finish off the bottom right corner
by realizing that R8C7 must be a 1. Finally, the unresolved 2 clue cannot go right, or R7C4 would be unfillable. This leads to this situation...
and the only way to resolve this (completing the puzzle) is
by an unclued size-5 region.