There is a balance with exercise and the immune system. At the right level exercise improves the immune system and will help fight viruses but too much exercise and the immune system suffers. As a result it's quite common for people to pick up colds after a big race, especially a marathon.
So in theory light exercise should help you fight your virus. I've heard you shouldn't run with a temperature though, as this places quite a strain on the body, especially the heart.
If you virus isn't settling though it's worth having a check up with your GP (if you haven't already).
It might be worth focussing your energy just on the running rather than doing cross training too. Stick to a slow pace and just do the endurance work (no speed or hill work) until you feel better. Maybe run 30secs to 1minute per mile slower than you'd expect for your distance work and see if that's easier on your body.