You should definitely be aiming to up your mileage and that should happen fairly naturally as you get fitter. I ran my first 10k off a similar weekly training load to you, perhaps a little more. I was typically doing 3 runs of circa 5 miles as hard as i could cope with and that got me through my first attempt but it was not a terribly scientific or sustainable approach.
I would personally recommend training with a heart rate monitor as that will help give you an idea of when you are fatigued from your cross-training and when you need a rest. Curiously its often pretty hard to tell if you are knackered simply based on how you feel and some objective measure helps.
A HRM isnt essential though as long as you keep an eye on how you are performing and dont try and push it. What i would suggest is slowly build up how far you run. A general guide is adding no more than 10% and even then dont do that every week, give yourself time to adapt and consolidate. If you do all your running at an easy conversational pace then you should be able to build the mileage up without too many issues but dont ignore fatigue or niggles. if in doubt, rest!
Definitely build one of your runs up to a long run but make sure its good and slow. The longer the better really, even for 10ks. Doing a good long run will build up your aerobic endurance. If you can do a 10 miler then 10k doesnt feel quite so hard. Again build it slowly.
Following those sort of principles, over the course of this year i have upped my weekly mileage from about 20 to over 40 on a good week and despite doing no speedwork at all i have taken 5 minutes off my 10k pb so high volume, low intensity is a mantra that can work.
