Seeds are sprouted. What now?

The bulk of my seeds have been sprouted for a week or so.  Some watermelons haven't come up yet as well as some peppers and tomatoes.  No big shock there.

I'm finding myself with a bit of a challenge, unfortunately.  We're still in the townhouse despite our perennial plan to get out "next year" (sigh!) with this being our 5th year of that plan.  I digress.

The challenge.  The Boy will be six years in a few months.  He's a very active 6 yr old and has Legos and whatnot all over his dresser where I used to put the seedlings until I was ready to transplant and put them on the grow shelves in the basement.  So there's no room in there.

Our daughter's room is kept dark since she's still taking naps.  She also is an exceptionally grabby 2 yrs old.  Far moreso than The Boy was at her age.

For now, we have the sprouts in the kitchen, but that is neither as warm as their bedrooms (which are south-facing) nor does it get all that much sunlight being on the north side of the house.  As a result, they're getting quite leggy. 

I could put them under the grow lights early, but that room in the basement just isn't as warm as elsewhere in the house.  Quite a quandary.