Log on to: http://www.ukvisas.gov.uk

I visited london before coming to US two years back. I took a visit visa from Nepal and flew to London. I am visiting again this September for a week before I go Nepal. There are three visa consulate office in US- Chicago, Los Angeles and Newyork. Just apply to nearest office through mail. Check in the embassy office for a list of documents. It shouldn't be that much difficult. But it also depends on  your status in US. I think that a student will be questioned more than a permanent resident of US. 

Good luck to you.

I applied through LA office by mail.
Firstly, you have to fill the online http://www.ukvisas.gov.uk (fee $137). Even though if you have a green card, you have to give the finger-print through your local uscis office (application support center). Appointment at finger-print center will be fixed at the end of the online application. After you have given you finger-print, you have to send your original passport, original green card (or original I-94), your bank statement, welcome letter from your relatives in uk  (air ticket & hotel reservation if you are travelling on business)  to uk consulate office through mail(bit scary,but it will be fine). So after 2-3 days, they will email you about the approval status. Then it takes couple of days to reach you.
I would say at least 15 days to get uk visa. 

see my attached image for business visa to uk