Often with sleep training, you will have to 'revisit' it at times, for example after your baby has been sick or you come back from holiday or they go through a developmental leap.
I would guess that even if you get your DD sleeping in her cot, you being in hospital is likely to unsettle her and there is a good chance you will have to re-do the sleep training when you come out of hospital.
Is it really practical for you to both leave the room in the night, especially if you are recovering from an operation? I would worry you will both become exhausted. It's one thing getting half woken up in night and staying in bed feeding baby back to sleep. It's another having to wake up properly and sit outside the bedroom then try to get back to sleep again. Imagine if you have to do that several times when you get out of hospital.
Is the medication you are on a short or long term thing? If short term and you would like to go back to co-sleeping (I only mention this as you have said a few times you love it) then you will be putting DD through CC for nothing if you are going to co-sleep long term anyway.
If you want a short term solution to DD sleeping in her cot, I would just muddle through as best you can. (If this is what you want then you will probably get lots of tips on how to do this).
If you want a long term solution then you may need to look at night weaning first, then finding another way for her to sleep.
If your aim is her not feeding through the night and sleeping in her cot in your room with you and DP in the bed, then there will be lots of good advice on here of how to best achieve that.
Your DD will probably be quite distressed having her primary carer in hospital (obviously you can't help this) but I do think there are ways to minimise the upset to her and avoiding CC is probably one of them.
She's going from breastfeeding/co-sleeping to being left on her own to cry, plus no milk through the night suddenly - if you add in ontop of that her mum going into hospital, it really is a lot to put a small baby through.
It does sound a difficult situation, especially when you explain DPs medication and hearing loss but there must be a better way.
Would the cot with the side down on DPs side not work? Surely he would hear/see her waking up then and could comfort/settle her without leaving her to cry.
If he's getting up to leave the room, he must be waking up.