I have found skink eggs here in S.Florida also but seemed they took longer than a month to hatch.

I just kept them in pure shade in screened patio in more or less same style set up as yours with container sitting in another container with water to prevent ants and misted them now and than .I found them late June last year and the summer heat was enough to hatch them.

I did not feed them when they hatched just let them go and though found all the eggs at same time under fallen log we were grinding up .
The eggs hatched at different days about 10 days apart. If you want to feed them they are tiny but still big enough to take pinhead crickets.

Yes we do need rain lol.