First of all, I think you made a slight math error. Every leap year, we don't end up short. We end up ahead. The leap year system assumes that a year is 365.25 days long whereas the year is actual year is 365.242198781. See, the actual year is shorter than the year as our calendar puts it. So leap years are actually putting us ahead.
To compensate for this, years divisible by 100 are not leap years. For example, the year 1100 wasn't a leap year. You calculated that we lose a day every 96 years. Since you got your subtraction backwards, what happens is we gain a day every 96 years. So after 100 years, we've gained a day and then some. So every hundred years, we don't have a leap year when we normally should (i.e. we lose a day) to compensate.
Some of you are probably saying, "Hey, wait a second! The year 2000 was a leap year!"
Others will be saying, "That's all very nice, but then won't we be losing a fraction of a day every 100 years?"
Both of these issues are resolved by a further wrinkle in the calendar. Years divisble by 100 are not leap years unless they are also divisible by 400. 2000 is divisible by 400, so the year 2000 was a leap year. The year 2100 will not be.
This still doesn't get things exactly right, but it's pretty darn good. At this point, we might be more concerned with such things as the fact that the rotation of the earth is slowing down (i.e. days are getting longer).
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