OP - I am really sorry you have suffered so much bereavement in such a short time. I can't begin to imagine how shocking this has been - and presumably continues to be - for you .... and how, amongst your own grief, you're also trying to keep it all together for your children, who'll also obviously be grieving.
Understandably, you want to protect them from further hurt - and to make their life the best you can. It's horrid to feel you might be letting your children down but, and I do mean this kindly, I suspect that everything you're going through has clouded your thinking. Perhaps not surprisingly - am speculating here, but maybe you've not been sleeping properly (?) and I expect you spend a great deal of time worrying how the future's now going to be without your husband, and how your children have been affected. What I'm trying to say - in the nicest possible way - is that it seems to me you can't see the wood for the trees right now, and that has led to you coming up with what you think is a simple 'solution' to your current worries.
Thing is though ..... it's not really reasonable, and whilst I think you're being driven by the desire to see your kids 'right', (rather than 'greed' per se), as everyone else has said, the most sensible and obvious answer to relieving your financial stress and improving the day to day quality of all your lives is to sell the rented house.
This would NOT be letting your children down in any way, shape or form. In one fell swoop, you could get rid of your debts - which will feel like a weight lifted from you - and, immediately benefit from the loan payments you're currently paying out remaining in your account - which means a much happier and less stressed mummy who'll be able to afford to provide the sort of activities and treats you feel your kids deserve and would benefit from. I don't know how large your debt is, nor what equity you'd take away from the rented house, but with any remaining capital you could use that to reduce your own residential mortgage - thus freeing up even more money monthly, or you could continue to pay that back at the same rate meaning it'd be paid off that much sooner. Alternatively, you could put that money away as the kids' university fund - which would be a fantastic thing to do for them and something which unfortunately many parents simply aren't able to do. Freeing up the capital from the rented house would give you so many options .... all of them beneficial to both you and the children one way or another.
I'm afraid I don't agree that BIL 'should' give you any of his money to solve your problems. As others have already said, it doesn't appear that he's done anything 'wrong' in this sad situation, and is therefore under no moral obligation to assist you. FWIW, in the vast majority of wills I personally know about, it's commonplace for the largest part, if not the total part, of the estate to be bequeathed to the next generation - i.e. their children - by parents ..... maybe there'd be a small token bequest for grandchildren, but this (IME anyway) tends to be a relatively small sum of money, or even more usually, things with sentimental value such as jewellery etc. I think most parents assume that what their children inherit will eventually trickle down to the grandchildren anyway, or, will benefit them in the here and now if their parents use their inheritance to improve the whole family's life NOW - which is what would happen if you use the rented house money. IMO, that'd be the most responsible thing you could do for your kids.
Again, I hope you don't feel offended if I suggest that you are perhaps looking at BIL and thinking 'it's not fair' ..... because he appears to have an easier life than you do. Hard fact is, we can all look at other people and think the same thing - but I appreciate when it's someone close to you, it's often hard not to feel envious. It's even harder not to feel envious of that person's 'easy life' if you think they don't 'deserve' it ..... am speculating, but you mentioned 'so much more horribleness' about him, so I'm wondering if you don't consider him a very nice person and don't have a particularly close relationship with him (e.g. your husband asking you specifically to keep in touch suggests he thought you might prefer to avoid him) ? If I'm right .... and you are thinking along the lines of him 'not deserving' what he has, then I do understand it must be hard to see him with, apparently, 'money to spare'. Thing is ..... he may well actually have 'money to spare', and he may well not be very nice, but all of that still doesn't mean he's obliged to you or your children. Maybe another thing to think of is that if he's a horrible person, then any money he did give you could potentially come with a 'price' ..... that could be as simple as him thinking of you as 'moneygrabbing' or 'greedy', or he could use the 'favour' he's done you in years to come as a stick to beat you with, never letting you forget and so on.
It's far far better to leave this alone so far as he's concerned. He's been 'lucky' to have received another substantial inheritance but that really is nothing to do with you. As others have said, he may have his own children at some point and they should really be the ones to benefit from his finances. One last thing to consider ..... if he's a horrible person, do you really want to encourage a relationship between him and the children anyway ? I know you made a deathbed promise to your husband, but I'm sure he'd not want you to be unhappy or for the children to be exposed to someone whose company isn't in their best interests. I'm not suggesting you tell BIL to bog off - which would be deliberate alienation .... you can remain open to contact (so long as he says or does nothing to upset you or the kids) but you don't necessarily have to encourage it. Let him approach you if he wants to see the children, keep in touch with birthday and Xmas cards (if he does the same) and maybe 'catch up' mails every so often but I don't think you'd be betraying your husband's wishes by not bending over backwards to force a relationship with BIL.
I'm so sorry for your situation, but I really do think the best answer right now is in your own hands and I think if you can resolve this yourself without depending on someone you don't seem to like very much, it will be so much more healthier for you, emotionally.