First the balance of the mortgage is immaterial to the taxable gain on the return.
Next the calculations of the gain are not as simple as you think. Mom should have been depreciating the rental property all the years it was a rental for profit. If no depreciation was properly taken then RUN to a local tax pro to get this issue corrected on the return ... this is NOT a DIY project.
yes she'll have to pay tax on the gain unless it was her personal residence for 2 out of 5 years before the sale.
then the amount of gain also depends on depreciation allowed or allowable while it was a rental.
A landlord will almost always owe capital gains tax, selling to a relative is not an exception.
If your mother is willing to sell below FMV to transfer the property to you, it might be possible to make it a gift, but there will be depreciation recapture to pay and if she doesn't pay it, you must, even if you live in the home as your main residence and will eventually qualify for the exclusion. But if your mom needs the money, you can't just "buy" the property below FMV and then "give" her some extra money, that's a sham and the IRS won't allow it.
You may want to take your situation to a professional tax advisor, tell them what you want to do and they can tell you the best way to do it.