If your card still mines along another card, don't do anything and keep mining. If it doesn't...
Run the card with another card that works. Use gpu-z to read out the broken gpu. If it's still recognized, you can try to flash a new bios onto that card with lower core values and maybe lower ram and see if it boots up again. An incorrectly set bios can be a reason for non-booting card.
If you want to flash the card, you will first need to read out the bios with gpu-z, save it and load the bios info with a program to change its values (forgot the name now), then save the new bios and flash it onto the gpu with Ati Winflash. I would use the original bios of the gpu, or worst case find another bios of the same card on the internet, download it and flash your gpu. All of this may help, ruin your card, or it may not help with reviving that gpu. If you have nothing to lose and know how it's done, it's worth a try.
Apart from this, the gpu is broken and can't be repaired easily. You could try to remove all plastic and put it in an oven for 30 minutes. Final straw. Read about 'baking gpus' in this case.
Make sure no pins or capacitors are broken or burned at the back of the card, even removing a gpu can cause stressed capacitors to fall off.
If none of all this works sell the gpu as spare-parts/defect on ebay.
Radeon Bios Editor