What to do about a student who missed multiple assignments and claims illness?
First, there may very well be policies in place to handle this, so I would check those first, in case your hands are tied in some way.

Generally speaking however, I'm fairly permissive with medical absences and helping students catch up, if (as it seems from your email) the desire is actually for them to catch up. In your case, for example, I'd schedule a meeting for the student to try and figure out a path forward.

I'm aware that this leaves me open to being exploited by the occasional student who has decided to fake an illness, but I'm far more concerned with make sure those students who do have major medical or life-related issues have the chance to do well in my classes, and I'd rather the occasional student get away with an undeserved extension than have someone who genuinely needed my help not get it.