I'm sorry, but if I'm (not OP) in the hospital for something serious requiring operations and a multi-week stay, responding to emails is somewhere around last on my todo list.
No need to apologize. I'm sure you understand how that necessary deprioritization might still look like ghosting or insincerity to others, especially someone who had just exerted significant effort on your behalf. If I had the ability to send one email, I'd hope to at least be able to send a second regretfully declining the offer.