Dear erlijyme -
As a US resident alien, you are taxed on your world wide income from whatever source derived, unless specifically excluded by the tax code. An outright sale would be included in your US income, but you would be entitled to a credit for UK taxes paid on the sale.
It might be possible to do a like-kind exchange (Section 1031) if the transaction hasn't closed yet. However, care must be taken in structuring a like-kind exchange and the services of a real estate attorney, professional real estate broker and/or a competent tax advisor to guide you through the process. I am providing a link to IRS Publication 544. See pages 11 - 18 for a detailed description of the various rules and steps that must be followed to meet the Section 1031 requirements. One of the requirements is that both properties be held for investment or productive use in a trade or business.
Renting your property to your niece at below market value rent, may mean that the property is probably "Not Rented for Profit". In this situation, you report the rental income on Line 21 of Form 1040, include the mortgage interest (as a second residence) and the real estate taxes on Schedule A. You can also deduct all other rental expenses, repairs, utilities, maintenance, etc on Schedule A as miscellaneous itemized deductions, subject to the 2% of Adjusted Gross Income floor. When combined with the mortgage interest and real estate taxes, these miscellaneous amounts cannot exceed the rent received. See page 17 of IRS Publication 527, and Page 5 of IRS Publication 535 , links below for deduction limitations.
If the "not rented for profit" scenario applies, the Section 1031 exchange provisions would not be allowed, and the gain would be included as taxable income.