For 2-digit numbers, Kaprekar's routine doesn't produce a constant in the same way as 3-digit or 4-digit numbers do. Instead, it leads to a repeating cycle of numbers. Specifically, for any 2-digit number with different digits, the routine will eventually fall into a loop: 45, 9, 81, 63, 27, 45, and so on. 

Here's how it works: 

Start with a two-digit number: For example, let's use 35.
Arrange digits in descending and ascending order: This gives you 53 and 35.
Subtract the smaller from the larger: 53 - 35 = 18.
Repeat: Now, use 18 as your new number. Arrange its digits (81 and 18) and subtract (81 - 18 = 63).
Continue the process: 63 becomes 63 - 36 = 27. Then 72 - 27 = 45.
The cycle: Finally, 54 - 45 = 9. And 9 becomes 90 - 09 = 81. Following this, you'll find 81 - 18 = 63. Then, 63 - 36 = 27, and lastly, 72 - 27 = 45.
Therefore, there is no single Kaprekar constant for 2-digit numbers, but rather a repeating cycle of {9, 81, 63, 27, 45}.