Is it a special overclocked model or stock AMD/ATI design ?
If you can boot with another card as the main graphics and the HD5770 as a secondary, you might be able to reflash the BIOS using ATIflash or another similar tool in a hope to get it working.
If you can't boot with another card plugged in, identify the flash IC (usually 8-pin chip at the back of the GPU), get an identical one, flash it with a fresh copy of the BIOS (make sure it's the exact model, or use the stock ATI one if you can't find it), replace the one on the card and cross fingers. You will need some sort of SPI flash programming interface, you can make one with an Arduino.
(The next generation (HD6xxx) got a dual BIOS feature where two copies of the BIOS were stored onboard and you could select which one to use for boot with a switch.)
If that doesn't work, completely remove heatsink and put the board inside the oven at 200C for 7 minutes then let it cool off for 15 minutes, reassemble and give it another shot.
If that doesn't work either, there isn't much that can be done.
Cheers,
Razvan
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