As said, if you can lock your suspension (if your bike has it!), then I'd do that and fit 26x1.5 Schwalbe City Jet tyres. When I switched to City Jets, my average speed on a 7 mile route dropped by over 17% and I think that you'll need to spend a fair amount to get a considerably better solution.
If you want to change bikes, though, do you like dropped handlebars? They are probably not a requirement over that distance and you are used to flats, so I'd stick with them and get a rigid (without suspension) flat-barred bike with 700c wheels, probably 32mm all-terrain tyres.
If you want to buy new and (relatively) cheap, my son has a
Specialized Sirrus which gets used for a similar route (about 4 miles to school) and he loves it. I think they are around £300 and I'd call that an ideal bike for your route. Note, though, that's
an ideal bike, not the ideal bike, as there are a lot of options from the likes of Carrera, Trek, etc.