She will have to return to the States and apply for her settlement visa there. It is not possible to apply to convert visitor visa into a settlement visa in the UK. Neither is it possible to get married in the UK when holding just a visitor visa.
But where will the marriage take place? If in the States then she will need to apply for a spouse visa after the marriage has taken place.
Alternatively, if the marriage is intended to take place in the UK, she will need to return to the States and apply for a fiancée visa, so that she can return to the UK and then proceed to get married here.
Not really bothered where the marriage will happen? In that case get married in the States and then apply for a spouse visa. Why? Doing that will cut out the hassle and expense of converting the fiancée visa into a spouse visa in the UK. The cost of that application would be £335 by post or £500 in person at a PEO ... Public Enquiry Office. It would also mean that as soon as she arrives in the UK on her spouse visa she would be able to work, whereas work is not permitted on a fiancée visa.
Lots more information on the
www.BritainUSA.com website, including how to submit an application online. Recommended! And then follow the instructions on screen about where to send the supporting evidence, where and when to attend for an interview, etc..
By the way, congratulations on the engagement!