Small petrol engine car is the only way for cheap miles. Only consider a TDI if your munching motorway miles.
My Litlle Hyundai i10 with its 5 doors and as much room inside it as my Ed30 is perfect. Cheap everything ,insurance, tax, £20 gets you 175 miles in return, ideal for the town and short trips. Nearly 55 mpg and no tax to pay..insurance the same as a packet of pork scratchings.
Only thing is you would need to IMO use your £2k as a deposit and get into a brand new car and have around £80-90 a month finance deal for around 4years. But that's up to you on terms etc.
You buy an old car, the have to add the maintanance costs to that...tyres, repairs, whereas a new car you don't.
Add to that a 5 year manufactures warranty, it's worth considering.They come with quite a good spec too as standard.
Some peeps might laugh, but seriously it's the best decision I made regarding choosing to run a 2nd car
HTH