some femal singers i enjoy:
neko case (someone mentioned her already - fox confessor.. is a great album)
nina nastasia
jenny lewis (from the band rilo killey, but also has a solo album out- Rabbit Fur Coat)
isobel cambell (formerly in belle and sebastian, but now doing solo stuff - her album with mark lanegan - "Ballad Of The Broken Seas" is worth checking out)
cat power
out of those the album most like what you are looking for (covers of electric songs) is probably:
'The Covers Record" by Cat Power - she has an unusual voice, and it wont appeal to everyone (be warned!!), but she does covers of all sorts of stuff like the rolling stones, moby grape and bob dylan, however they are often unrecognisable
somethings that are bit more mainstream (but with their own material, not covers):
Neko Case - Fox Confessor Brings The Flood
Nina Nastasia - On Leaving
Jenny Lewis and The Watson Twins - Rabbit Fur Coat
are all fantastic albums