I agree with others: this is not a real friendship, but at best, a professional relationship.
It's clear she contacts you only when suitable for her, either to use you, or to show off in some professional capacity. Which is pretty normal for professional relationships, but highly abnormal for an actual friendship.
I'd mentally change your 'relationship status' with her and simply respond in kind. You two have professional contact, great, keep it that way. Who knows--she might be useful to you one day as a reference again or something. But don't expect her to actually care if you're unwell. She does not see you as a friend and does not care about you beyond a strictly professional capacity.
If realising this isn't too off-putting for you, then respond to her note. Ignore her excuse/blatant lie and just congratulate her on her promotion. Continue to contact and communicate with her in only a professional capacity. Although that said, personally I'd be too upset and hurt to do this.
Realising someone doesn't care about you sucks. If you wanted to cut off from her, just stop contacting her. It seems like she'd have no qualms about doing the same to you, since she already did that at the exact worst possible time. Maybe send a "congratulations on your promotion. I'm busy recovering so I won't have time to contact you for a while. Best wishes" note if you're worried she won't get the point.
Concentrate on getting better with the people who actually care about you. You need their love, not hers.