First of all OP - and this is going to sound weird - but are you sure it's your knees? Often the pain is felt in the knees, but issues are actually caused by the illiotibial band (google ITB pain). The pain is felt where the band of fibre, that comes down the outside of your leg, meets the knee, and presents as "knee pain", but is actually brought on by weak glutes not doing their job.
If it is the ITB, rather than the knees themselves, and if you catch it early enough, some foam rolling and glute strengthening (bridges, clam shells) can work wonders.
Next up, trainers - forget gait analysis for now. As a new runner, 99% of the time a gait analysis would tell you you overpronate, you land too heavily on your heel, and you need support shoes. Instead, what you need to do is get used to running and find your natural form, before adapting your shoes. A lot of that comes with speed, with practise and with losing weight (simply because, say, 9st hitting the ground is always going to be less clompy than 13st hitting the ground).
So don't worry about your gait or your foot strike too much for now. There's not much you can do to correct your foot strike anyway - just do what comes naturally, while concentrating on keeping your head up (as if an invisible string is coming out of the top of your head), running lightly on the ground and aiming your heels back towards your bum.
From personal experience, as a 12st 30-min 5ker I overpronated and had ever-painful legs and knees, whereas as a 9st, 20min 5ker now, my form is much better and I'm not injured as often. And my form got much better when I ditched the support shoes I'd been talked into!
The fact your knee pain is already improving makes me think it's just your body adapting, rather than a real "injury." So think about getting strong all round. On youtube there are some great free strength and conditioning resources for runners. Simple stuff like strengthening your glutes, core, and quads will really help make sure your joints aren't taking all the strain.
Good luck with it and do let us know how you get on! Oh, and treat yourself to new shoes when you graduate - something neutral and cushioned 