What should I do about the Ramadan fasting I missed due to illness, given that the doctor ordered me to break my fast for two consecutive years, and I have fasted one month, with one month remaining after other Ramadan months have begun?

You are not at fault for breaking your fast due to illness, but you must make up the fast after your recovery, before the next Ramadan (definition). If the making up of the fast is delayed until another Ramadan, then in addition to making it up, you are obligated to offer expiation by feeding one poor person for each day. If you have made up the fast of the first year, and only the fast of the second year remains, and you fasted it before the next Ramadan, then you are absolved.