'Hello' is an excellent all purpose greeting. It is completely meaningless except for its function as a greeting, which makes it a little bland but also means that it is extremely unlikely that it will draw a negative reaction from the recipient.
'Sincerely' is an old classic sign off for letters, and it works well here because it doesn't contain any optimism or well wishing that the receiver might feel is misplaced. It does nothing to remind the receiver of their unfortunate state of health, instead merely emphasizing that you truly mean the words you have written.
This sign off might come off as a little formal because of its age and history, but you've indicated that your relationship is semi-formal in nature so I don't think that 'Sincerely' will feel too out of place.
Whatever you were using before
If you have been frequently communicating with this person using a small set of greetings and sign offs, I recommend you stick with what you were previously using. A change in your greeting, no matter what you change it to, will only draw attention to why you are changing it, which is exactly the opposite of what you desire. Greetings and sign offs are mostly filler anyways, and rarely do people ascribe their full meaning to them.
That said, this advice only applies if your communications are frequent and consistent enough that your correspondent will notice the change. If you think that the change would pass uncommented, then by all means find a more neutral set of phrases.
Different people react differently
Each person reacts differently to bad news. Some people will welcome and be encouraged by well wishes even if the health being wished for is impossible. Others will be frustrated by the same greetings. Figuring out which sort of person your correspondent is will help you formulate the best greetings for them.