trying to stay in the UK - student visa? other ideas?

I am a US citizen here on a visit visa (about 6 weeks into the 6 month term). I am living with my boyfriend (non-EU citizen with ILR status in the UK). I am trying to figure out how I can stay here longer than 6 months.

I haven't broken any immigration rules. I am trying to find employment and get a work permit sponsored. I have a BA and some skills, and recruiters have told me if I search for several months I may find someone willing to sponsor me. I am trying. But I am not overly optimistic, as the market is flooded with UK citizens, EU citizens, working holiday makers, people here on heritage visas, etc. I am sending my CV all over but companies simply don't sponsor work permits period.

I can't get a heritage visa. I can't get HSMP (only 50 points). I can't get a domestic partner visa (haven't been with my boyfriend nearly long enough). I have tried to look into all my options but I could really use some advice.

I am looking into the idea of going to school here and staying on a student visa. My research turned up a lot of very expensive programs (especially for international students) and a lot of dirt cheap English schools. Are there any somewhat more affordable programs that would qualify me as a full time student (I think the minimum is 15 hours a week of study) in something other than English? I think a native English speaker signing up for English lessons would raise some eyebrows at immigration....

Anyway, I am not here to try to beat the system. I don't want to break any laws or work illegally. I would be perfectly happy to work part time as a student visa allows (or full time if someone would sponsor me!), but I really can't afford 8000 or 10,000 pounds to go to school right now. Does anyone know of inexpensive schools that can qualify me for a student visa, or have any other ideas for someone in my situation? I am interested in a wide range of subjects and I have some money to put toward education, within reason.

Also, let's say that I can't figure out a way to get a visa within six months and my visit visa runs out. How likely is it that they will let me in for another six months if I go back to the US and then turn around and come back to the UK 4 weeks later? Technically there is no rule against visiting twice in a row (right?) but I have a feeling getting another stamp so soon might not be easy....

See also 10.2 Requirements for a visitor under the Rules (Frequency and duration of Visits) and Passport Control at Heathrow and Exploring the decision making of Immigration Officers.

Try contacting one of the American colleges?

How about accounting? The fees aren't as much as a university and you do get a valuable qualification at the end.

Hi - there is no absolute rule about how long you are supposed to be here on a visitor's visa, but the BIA use a rule of thumb that you shouldn't spend more than 6 months out of any 12 months here.