You could probably get away with even a cheap 15 quid multimeter or perhaps even cheaper. I could also probably quite easily guide you through taking core and memory resistance after that though when you know the core is good/bad what next? Your likely going to be tracking a short and will have to probably replace at least one smd component. To find the short you will need a desktop power supply, then you need the right tools to replace the smd. Some spare GPU's to borrow parts off of etc etc etc...
I would give myself a less than 1% chance of successfully replacing a component, it's the kind of thing i'm just not good at.
I've found this company - http://www.pcfixltd.co.uk/gpu-repairs - but i've no idea how competent they are or whether or not they even attempt these types of repair.