I think that it may be mostly separate.
Lack of sleep is correlated with being sick, and if your LO is now waking you for more feeds then that is a bit of a nasty cycle.

But the number of nasty viruses going around at the moment is horrendous - we have also had what seems like a non-stop run (courtesy of DS collecting them at childcare/swimming/kindy; he gets somewhat sick and DH and I get somewhat sicker). And we really can't blame the few feeds he has a week for making us run down.

If anything your immune system should work better whilst breastfeeding, so stopping now may not be very good for getting rid of the virus.
And I am a bit concerned that by blaming BF-ing you may be missing another more important issue.

Can you make your own nutrition, gentle exercise and sleep your priorities, without full on weaning? Try feeding lying down (on the floor if you have to) so that you don't have to hold miss M, and so that ideally she can come to you. And can your partner take on more night duty (armed with bottles if need be) for a little while?

I hope things improve.