What should I do, as I swore, "I will not do that thing, and if I do that thing, I will fast ten days," then I broke the oath and did not fast the ten days?

If you swore by Allah not to do a specific action and then broke your oath, you must offer expiation for the oath: feeding ten poor persons, or clothing them, or freeing a slave. If you cannot find any of these, then fasting for three days.

As for fasting ten days as a commitment from you, not as an act of drawing closer to Allah, this is an oath of contention (nadhr al-lajaj). In this case, the one who made the vow is given a choice between fulfilling the vow or offering expiation for an oath.

And if you said, "If I do such and such, I will fast ten days," without swearing by Allah, this is also an oath of contention, and it involves the aforementioned disagreement among scholars.