Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear questioner, thank you very much for having confidence in us, and we hope our efforts, which are purely for Allah's Sake, meet your expectations.
There is no expiation to be made by the legally responsible person who has deliberately not observed fast in Ramadan. He is to make up for the fasts missed as a means of showing sincere repentance to Almighty Allah, [not as a kind of expiation.] Expiation by fasting two consecutive months is obligatory on the one who has broken a fast by making love to his wife.
This does not mean that one who has deliberately not fasted in Ramadan has committed a sin less than the one who has intended to fast a day and then broken it by yielding to his sexual desire. The latter is, rather, like one who has sworn an oath not to do something but then has done it, while the former is like one who has initially sworn a false oath.
Neglecting fast in Ramadan on purpose and swearing a false oath are both major sins for which there is no expiation. A person who commits a sin of such a kind is to repent sincerely and with deep regret to Almighty Allah, so that He Most High may forgive him.
Responding to your question, the prominent Muslim scholar Sheikh Yusuf Al-Qaradawi states the following,
I am not going to reproach you, brother, for neglecting your duty towards Allah Almighty with regard to fasting Ramadan, which is one of the greatest obligations of Islam, for a whole ten years. Your conscience has done you enough in that respect. Praise be to Almighty Allah that He Most High has guided you to the right path after you had deviated from it all that time.
We may hear that many Muslim men and women observe the fast of Ramadan, but, unfortunately, neglect to perform Prayer. But you, brother, have reversed the matter; performing Prayer but neglecting the fast of Ramadan. Almighty Allah has made fasting that month each year obligatory on legally responsible Muslims, so that they train themselves to ward off evil. Allah Almighty says, “O you who believe! Fasting is prescribed for you, even as it was prescribed for those before you, that ye may ward off evil.) (Al-Baqarah 2:183)
You, brother, have not denied the obligation of fasting Ramadan. You gave up fasting on purpose, yet you remained believing that fasting is the fourth pillar of Islam that is obligatory on every legally responsible Muslim. Furthermore, you have been keen along these years on performing the obligatory Prayers.
Hence, we cannot say that (by your giving up the sin and returning to fast) you are like the person who has given up disbelief and followed the right path of Islam and thus all his past sins are forgiven. Along these years, you have been sinful by neglecting your duty of fasting.
As for your asking about what you can do to redeem such a sin, you are required to make up for the fast days missed in those years. This is like a debt that you have to pay sooner or later. One's duties towards Almighty Allah and people are like debts that one owes to them. In that regard, there is an authentic hadith to the effect that one's duties towards Allah Most High are to be given priority in fulfilling.
The elapse of years does not exempt one from performing one's duties towards Almighty Allah and people. One's duties remain as unperformed obligations upon one until one performs them or bears the consequences of not performing them on the Day of Judgment.
To sum up, dear brother, you are required only to make up for the missed fast days. That is to say, you are required to fast three hundred days to make up for the fast days missed in those years.
You are not required to fast, as you say, sixty consecutive days for each day missed. If that were required from you, it would mean that you would fast eighteen thousand days!
Some people think that the legally responsible person who has not fasted Ramadan on purpose is to expiate this sin by fasting sixty consecutive days for each day missed, as in the case of one who invalidates his fast by making love to his wife.
Mind that there are some schools of jurisprudence who believe that this expiation is confined to one who breaks his fast by sexual intercourse, while some other schools are of the opinion that it applies to one who invalidates his fast by making love and also by eating.
The expiation in that regard consists of freeing a sound Muslim slave, or if not possible, then to fast the days of two consecutive months. If this is not possible, then the expiation is to feed sixty needy persons for each fast-day invalidated.
But this is an expiation imposed on one who has principally intended to fast and then invalidated it. It is imposed as a purification for him from the sin that he did not premeditate.
This expiation does not apply to one who initially did not intend to fast (or, in other words, who premeditated not to fast). One who did not intend to fast is like one who has sworn a false oath by Almighty Allah. [Expiation is not imposed in both these cases, for there is no expiation, however great, that is tantamount to the gravity of the sin involved.] The only thing the sinner can do in these cases is to express deep regret and submissively repent to Almighty Allah.
Repentance is available in all instances of sin, whether minor or major. But in order for repentance to be true and acceptable, one is required to redeem one’s sins by reforming the mistakes one has made and giving the rights one has violated back to their owners. This is a condition for accepting repentance by Almighty Allah.
(Hence, you are required to make up for the fast days you missed, not as an expiation, but as an expression of true repentance.)
Mind that it will not be acceptable to feed a needy person instead of fasting a day to make up for a fast day missed. This can be done only when one is not able to fast at all. But if one is able to fast, nothing but fasting will be accepted from one.
I advise you, dear questioner, to make up for the fast days missed during winter when the daytime is short and cold, so you will not get so exhausted during fasting.
It is said that winter is such a good time for believers. Its days are short and, so, they can fast them (without great exhaustion), and its nights are long, which will make them enjoy offering the Night Prayers.
Hence, you can, dear questioner, fast three consecutive months each winter (until you make up for the ten months missed). I assure you that if you do so consecutively, you will not find it difficult, especially if you do so with a deep belief that you are redeeming the sin you have committed and pleasing your Almighty Lord.
May Almighty Allah help and strengthen you in what you are going to do.
Allah Almighty knows best.