Gaining US right of residency as a British Citizen?

There are no partner immigrant visas, only spousal immigrant visas. Unless you're willing to marry her, she has no means to qualify for a green card. She will need to obtain a non-immigrant visa to live with you in the US. Employment visas are very hard to get, limited mostly to skilled jobs in science and technology. If she can afford to pay international tuition, she could apply for a student visa.