I dont think that you can resurect u could give her 60 more seconds of life back with a
zombie tomb spell.

If you have a previous save (unlike me) then load it and give your housecarl better
weapons and armor. While it sucks reloading housecarls can be good with the right tools
and cheap followers.
Not to mention they are awesome walking talking armor stands that way.

I believe you can go back to the jarl of whichever city you obtained the housecarl and ask
for another one.

If not, load a previous save if possible.