You'll have to spend days reading the confusing information on the UKBA website, and you'll probably find at the end of it the only way is to get married. There are lots of different types of visas that may be applicable depending on her circumstances.
But there's a good chance none of them will be and you won't be able to do it. It's a stupid situation when you consider we have hundreds of thousands of illegal immigrants who come here on the back of lorrys but we can't deport them, but someone who came here legally and is in a relationship with a Brit can't stay.
It may have changed since I went through it all a few years ago, but at that time, the only way to get a visa on the basis of being in a relationship with somebody British or with right to reside here, is if you've lived with that person for 2 years and can prove it or if you're married to them.
Of course there are other visas you can look at that are nothing to do with partners. Student visas are probably the easiest to get, although still not that easy and you do genuinely need to be studying full time.
There are visas for people in certain occupations that are deemed to be in short supply, so if your partner does something that's on that list, you can look into that.
As she is a commonwealth citizen, if she has some British ancestry, there is a visa she might be eligible for.
The best thing is to just start reading up on all the visas available.
