Poor you. Click on this link to read an excellent article about the links between depression and infidelity. I'm pretty sure you will see your H in those pages.
I find it interesting that he got a clinical diagnosis in September and he started this affair at around the same time. Often in these situations, it's a puzzle about which came first, the depression or the infidelity.
What I have learned is that depressed people are much more vulnerable to affairs, because the affair mimics a chemical high. That "high" becomes much more addictive and intoxicating to the person who has flatlined. To a depressed person in mid-life, the feelings of "being alive" brought on by an affair are so powerful, that they contribute to some very poor life decisions.
In your case, you've got a lot of cards stacked against you. You've got a depressed man in mid-life, having an affair with an old flame. Affairs with old flames are recognised to be the most dangerous to the primary relationship, because the lure of them is not just the person, but the memories of being young and without responsibility; another push factor in mid-life.
It sounds as though you have been understanding and benovolent so far, but although it might not seem like it now, the discovery of the affair will liberate you.
Now that you've got your discovery, recognise that a large portion of the behaviour you've been seeing has got nothing to do with the depression and is more associated with the affair. If you want to read about this, concentrate your reading more on mid-life affairs - and less on depression.
He is evidently biding his time, to leave you when it suits him and his OW. Just because he is depressed, doesn't mean he is not hugely manipulative and at the moment, he is trading on your humanity to continue giving him a roof over his head.
Call his bluff now and ask him to leave. This relationship with the OW needs to stand on its own four feet now in real-life, away from the romanticism of secrecy and plotting.
They are doing this on your time, trading on your sympathy for a loved one who is "ill" - and that should make you very angry. Use that anger wisely and tell him the game is up.