This situation is actually fairly simple. The nice part about your question is that popularity of a given answer doesn't matter. As of this moment, all respondents on the poll voted for paying your loans, and mathematically speaking all of them are utterly wrong. Since you stated that you want to maximize the financial returns from your windfall, you are asking the following question :
between these three choices, which maximizes the expected value of my future wealth?
Well, with IBR (Income based repayment) and with PSLF (public service loan forgivness) it makes 0 financial sense to pay any of this money towards your loans. As I have said in other threads, it is a bad choice on the part of the federal government to have instituted this program, but the government cannot renege on a written contract except in circumstances where national security is at stake. Thus, it is in your best interest to NOT pay a penny beyond the minimum payments required under IBR (hopefully this windfall showed up BEFORE the tax year being considered for IBR payments) because that maximizes the amount of money you will have forgiven when you receive public service loan forgiveness. To receive PSLF, you merely have to complete residency while making IBR payments and then work at least 30 hours a week directly for a nonprofit (like nearly all hospitals) until 10 years have passed, again making IBR payments. After that, zap, your debt is gone - and any money you paid now from your windfall would have been totally wasted.
As for the rest of it, this is also trivial. Allocate your windfall between your new house and your old house, keeping in mind the interest rates and incentives for making a large downpayment. For example, if there were no incentives, then you should make the minimum downpayment on your new house and put the rest towards the mortgage on the old one. However, I bet if you make a 20% downpayment on the new house, that will substantially reduce certain costs, and so it's worth more than paying down that 6.25% loan. Find out if there are any incentives for making an even larger downpayment, since the larger the payment, the less the risk the bank is exposed to for making the loan.
One quick note for the uninformed : expected value = probability of event A * financial change of event A + probability of event B * financial change of event B and so forth. This is important, because IBR/PSLF might not pay off like the government is currently promising. But, under this 'windfall situation', this is trivial. Assume there's a 90% chance of the government paying the debt under PSLF, and a 10% chance of it reneging. You choose to invest by paying down the bank loans for a guaranteed annual return of about 6% (a rough estimate, since you've got 2 loans to split the money up between). Then the expected value = .9*(~6% compounded return on investment over 10 years * amount of windfall) + .1*(6-6.8% compounded * additional student loan balance you would have paid with amount of windfall). I hope you can understand my math, but the interesting part is that in the worst case scenario where the government reneges, you lose 0.8% * windfall with some compounding over 10 years, or about 15% of your windfall.
However, if you dump all your money into the student loan balance, you lose 100% of that money if the government doesn't renege. Thus, there would need to be a very high chance of the government reneging (more than about 80%, but I don't feel like coming up with exact numbers right now) for it to make financial sense to invest in your students loans instead of your house. I don't trust the government's motives and I believe it makes one colossal screwup after another, but the historical record shows that the government is probably going to pay out on the PSLF 'entitlement' program, and it would be stupid to bet against it.
