What should I do if a professor is not writing the recommendation letter whom I already suggested?
Use a different referee, time is now not on your side and she did not plan to be sick.

If she is not capable of replying to emails she might be very sick so have some respect.

You might consider sending her reference late with a note of explanation but I suggest you send the initial application fully complete. Many times the first stage of the sorting process is « is it complete » and only those that are get to the next round.