Looks right, given some assumptions. Let's build it up a different way to check your work.
We're basically interested in two patterns of putting kids into rooms. Those are all kids paired with one empty, 2-2-2-0, and two pairs and two solo sleepers, 2-2-1-1. Any other arrangement either breaks the 2 kids max rule or is a permutation of one of these, like 2-1-1-2 or 2-0-2-2. Permutations are easy so let's ignore that for now.
Let's start by figuring out how many unique sets of pairs of kids we can have. There are 15 possibilities for a pair drawn from 6 kids, lettered A-F: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, and EF. We might now be tempted to take one pair, then figure out the number of combinations for pairs of the remaining four kids for the second pair, and the last two are the third pair. That would give us $C^6_2*C^4_2*C^2_2= 90$ sets of kids.
However, we're double-counting; we could have chosen AB for the first pair and then CD for the second, or vice-versa, and our current calculation assumes AB-CD-EF is different from CD-AB-EF. It's not, so the figure of 90 is $P^3_2 = 6$ times too high; there are actually only 15 unique sets of pairs of all 6 kids that can be formed.
Now, of those 15 sets of pairs, you can either just chuck 'em into rooms, leaving one empty, or you can split any one of the three pairs to form a unique 2-2-1-1 arrangement. So, there are just 60 ways (15 + 3*15) to put 6 kids into 4 rooms with a max of two per room if you don't care who gets what room. If there's an unwritten understanding of one child minimum, no empty rooms, then there are 45 ways just counting the unique 2-2-1-1 arrangements.
Now, if you do care who goes in which room, then there are 24 permutations of 4 things taken 4 at a time. 60 unique ways to split up 6 kids up into 3 or 4 groups, times 24 ways to put those kids into 4 rooms, gives you 1440 unique permutations of kids in rooms if an empty room is allowed. If there must be at least 1 kid per room, then 45 unique 2-2-1-1 patterns, times the same 24 permutations, gives 1080 ways to put kids into specific rooms with at least 1 in each.