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First, you will not count any amount that was previously expensed as repairs or taken as depreciation. If taken as an expense of the rental you can not count that again as part of your cost. So, those improvements will be added but then the depreciation expense will reduce the amount of those improvements that add to your cost basis.
Purchase + improvements(not deducted as repair) - depreciation = cost basis
Second, you can not claim as part of the basis sold any amount that was paid for the portion of the property you will not sell.
Third, the amount of depreciation that was taken will be at ordinary and not capital gain rates (but given this was rental property will be at a maximum of 25%)
You do need to separate the cost of the property you will retain from the property being sold. If you do not sell the kitchen then that improvement is only added to the cost of the portion not being sold.
You can use any reasonable method as there is no particular method required by the regulations to divide the cost of the sold and not sold portions.
For example, if the assessed value had 100K for the building and 250K for the entire property it would be reasonable to use the same ratio and apply it to your original cost. That would mean that 100/250 times 63K or 25.2K was your cost of the building and the other 37.8K was for the rest of the property.
Use those prorated amounts to separate into the sold and not sold portions and then add improvements and reduce by depreciation whatever applies to each portion.
Keep the computations of the unsold portion for future use when eventually sold.
If there are items for which you have no knowledge a local realtor may be able to give an estimate of value (such as for "driveway access")
Please be aware that the assessed value for property tax is almost always less than the fair market or current sale value. Paying the assessed value would be a very good price in almost all tax jurisdictions. My jurisdiction shows comparable sales and those sales prices are about 20-25% higher than assessed value of the property.
Please do ask if you need more discussion or clarification.