I just thought I'd share as I was looking to build something that would be able to play any media format with ease, plus later I wanted to be able to leave it encoding video's (tidying up my media server) this build thread got me thinking (and spending), but given what your requirements are, I'm sure you could do this for well under £200!
CPU - I only considered IvyBridge w/IntelHD4000 integrated GPU's- seemed to be the sweet-spot for performance and price, especially for RAM+motherboards!
i3 3225 (£45 inc P&P, eBay)
or if you're near Crawley: i5 3335S w/IntelHD4000 integrated GPU (£55 eBay, but local pickup only)
Mobo - Asus P8H61-I/RM/SI (£39.95 eBay)
RAM - 2x8Gb Kingston Ultra Low Profile1333Mhz (£59 eBay) - overkill, but 8Gb was mostly ~£40!
Cooler - Thermaltake Gravity I1 (£13.50 Amazon)
Antec ISK 300-150 mini ITX case (lots ~£50 eBay)
I had the advantage of a 240Gb Crucial MX200 SSD + Windows 10 already, but I'll dual boot it with a USB3 Linux stick anyway (64Gb for £15 on Amazon), so you could just run Kodibuntu/similar on a USB stick....
Everything else is a fresh build and priced up today.
Based on the build that I'm just doing, but going for a slightly lower spec CPU (but still HD4000 GPU) and less RAM, you could make a budget Kodi build for <£180!!
£45.00 CPU i3 3225 (eBay)
£13.50 cooler Thermaltake Gravity I1 (Amazon)
£40.00 mobo Asus P8H61-I (eBay)
£16.00 RAM 2x2Gb (eBay)
£50.00 case Antec ISK 300-150(incl.PSU) (eBay)
£15.00 OS drive USB3 64Gb (or ~£40 for a SSD?) (Amazon)
£179.50 total, inc P&P.
If you wanted to spend the full budget, you could spend an extra £22 on the RAM (£38 for 2x4Gb Kingston DDR3, eBay) and an extra £32 (£47 for a 120Gb Samsung EVO750).
That takes you up to £233.50.
Honestly, I'd do the 8Gb RAM and leave out the SSD for now (£201.50 total).
If you're worried about CPU performance: the i3 above is comparable to the AMD 5350, but the i5 3335 (approx £10 extra) absolutely leaves the AMD standing!
http://cpuboss.com/cpus/Intel-Core-i5-3335S-vs-AMD-Athlon-5350