When your mother quit-claimed the house (gave you the house) she also gave you her cost basis. You seem to have accounted for that correctly. However, your "adjusted tax base" is irrelevant, because the home may not be assessed at full value and items added to the tax base may or may not count towards the cost basis. You need to know the adjusted cost basis. The calculation is similar, but the difference is important. Your mother also gifted you her holding period, so the sale is treated as a long-term capital gain.
https://www.irs.gov/faqs/capital-gains-losses-and-sale-of-home/property-basis-sale-of-home-etc/prope...
Not all closing costs are allowed as adjustments to basis. See IRS publication 523 starting on page 8 for a list of allowable adjustments.
https://www.irs.gov/pub/irs-pdf/p523.pdf
However, it's not clear you qualify for a partial capital gains exclusion. What is the unforeseeable financial hardship that forced you to sell before the two years was up? I will assume your mother gave you the home knowing she was sick, and you came to live with her and care for her, and then 8 months later she needed residential care. For you to claim a partial exclusion, you must meet one of the safe harbors, or you must show by other facts and circumstances that you had significant financial difficulties in keeping the home, and these difficulties were unforeseeable.
(The safe harbors are death of a spouse or co-owner, divorce or separation, birth of twins, and loss of your job; none of which seem to apply.)
What facts and circumstances made you unable to live in the home after your mother went into residential care? Not just a preference to move, but unable to remain in the home? The fact that your mother needed care would have allowed her to sell, but she wasn't the owner, you were, and the rule applies to you, not her. You need to show that you needed to sell due to an unforeseeable hardship. And even if a hardship did exist, was it really unforeseeable? It seems most probable that you could have reasonably foreseen your mother needing residential care and you wanting to sell and move.
So without more from you, I don't think you qualify for the partial exclusion.
For more, see publication 523 beginning on page 6.
Real property is land plus anything permanently attached. If the mobile home is not permanently attached, then it is separate tangible personal property, and you either sold it separately, or it was included in the contract for a nominal $1, or something like that. (Similar to how if you sold the home with appliances, the appliances are not part of the real property and were technically sold for $1 along with the real property.)
Your tax calculation is incorrect. I can't tell where you got your numbers.
Your numbers show an adjusted cost basis of $148,713--cost basis plus improvements that are permanently attached to the real property. I am ignoring the mobile home, and if the lean-to, awning or cabana are not permanently attached to the real property, they have to be taken off. (For example, an awning erected over a parking space to create a covered car port would not be a property improvement since it can be removed.)
You have a selling price of $285,000 less selling expenses of $23,676 (assuming all the costs you included are allowable).
That gives a capital gain of $112,611. That long-term gain is taxed (federally) at either 0%, 15% or 20%, depending on your filing status and your other income. Without more information from you, I can't say, but most people pay 15%.
Most states don't have a separate capital gains rate so this will be added to your regular state income, state tax rates are between 3%-13% depending on the state and your income.
If you did qualify for a partial exclusion, then your taxable capital gain would be $112,611-$83,333 = $29,278, which again, is taxed at 0%, 15% or 20% depending on your other income.
