It's true the Mayan "long count" used a tun of 360 days, but it doesn't matter: It wasn't intended to be a solar calendar so it doesn't need to sync up with a solar year or our calendar. It was used simply to count the number of days since the "beginning of time" and 188.8.131.52.0 in the Mayan calendar is December 21, 2012 in ours (assuming we've got the starting date correct), regardless of our leap years.
However, the date of December 21, 2012, is not based on Mayan tuns. It is based on the total accumulation of days since the beginning of the Mayan calendar. It is commonly accepted that the first date on the Mayan calendar is August 11, 3113 BCE on the Gregorian calendar. Therefore, we must start on that date and count forward in time.
The Mayan baktun is the equivalent of 400 Mayan tuns (years). But remember, their years are only 360 days long. 400 x 360 = 144,000. Therefore, each Mayan baktun is a total of 144,000 days long.
As of the time of posting, the Mayan date is 184.108.40.206.11. The first number represents the baktun, the second number katuns (20 Mayan years), the third number is tuns (Mayan years), the fourth is uinals (20 day “weeks”), and the final number is the day. I know that I’m making some history nut cringe at my over simplification of the Mayan calendar; I’m using “years” and “weeks” to make things easy to understand.
On December 21, 2012, the first number in the Mayan calendar will change from 12 to 13, making it 220.127.116.11.0. The last time this happened was on September 18, 1618, when the current baktun started. As you have already imagined, the Mayan date on that day was 18.104.22.168.0. Obviously, the world didn’t end.
But how do we know that 22.214.171.124.0 will happen on December 21, 2012? First, we need to calculate the total amount of days required to reach the 13th baktun: 144,000 x 13 = 1,872,000 days.
In our Gregorian calendar, 1,872,000 days since August 11, 3113 BCE is December 21, 2012.