Is it obligatory to say the Adhan and Iqama, or only the Iqama, for prayer at home due to illness?

What you are doing is correct. The Adhan (call to prayer) and Iqama (second call to prayer) are both recommended for an individual praying alone, due to their general virtue, and his prayer is valid without them. It is more complete to make the Adhan, as there is great virtue in it. It is prescribed for an individual, whether he is in a desert or a town, and whether he hears the muezzins around him or not.