I’m sorry to say that I think it’s better to let go and try again next year.
I do know how exciting it can be to finally attain Status for the first time (it was for me!). And if you were only 20 Status Credits away from Gold, it may well be worth chasing. But Silver simply isn’t worth the extra money you’ll have to spend on either a “status run” or a credit card that comes with Status Credits.
My suggestion would be that you either try again next year, or you could consider taking out Qantas Club membership if it’s lounge access that you’re after. If you go for the Qantas Club membership, be aware that there are many unions, professional associations, clubs and businesses out there that have discounted QC membership as a perk – you might be associated with one already.
Re your idea about asking Qantas to postpone your deadline for medical/health reasons: it’s always worth asking nicely. The worst thing that can happen is that they’ll say “no”, but you might be pleasantly surprised. Make sure you let them know that you can produce a medical certificate if they want one: that will help them to know that you’re genuine.
Hope this helps and all the best for your recovery.