How does one make up for the missed days of Ramadan due to a legitimate Sharia-recognized excuse, which have accumulated over several years, making it difficult to determine their exact number?

You must adhere to God's law and distinguish between what is lawful (halal (definition)) and unlawful (haram (definition)) through knowledge. Questions about matters should be asked before acting upon them. You must repent for any negligence in asking people of knowledge, and repent for delaying fasting without a valid excuse.

After repentance, you must make up the missed days and pay an expiation for delaying the fast (feeding a poor person for each day) if the delay was without a valid excuse until the next Ramadan (definition) began. However, if the delay was due to a valid excuse, such as illness, then you only need to make up the days without expiation or sin.