Yes,
AFAIK the Cooler Master eXtreme Power 600W should be able to handle the HD 5770.
At ~10K, nothing beats the HD 5770 at this moment. If you do go for the HD 5770, you should note that in the range of 9.5K to 10.4K you might find a variety of brands of the HD 5770. Some *are* better than the others, so don't just try to get the cheapest one.
For example, the Biostar HD 5770s are really not good for heavy gaming use - the heatsink-fan combo in that card is cheap and hence will not cool it as well as a good brand's card would (however, the Biostar is a LOT cheaper than other HD 5770s).
Some good brands you should be looking for are PowerColor, VTX3D, Asus, Sapphire and HIS. XFX is also good but recent cards of theirs have had some issues (if you have any interest in overclocking the graphics card btw, don't get an XFX card!).