It's a fair bit of effort to code this, so I doubt anyone is going to hand you the solution unless they are REALLY interested in the problem.
Let's simplify it- forget the part about the sum of the digits, the harder part is calculating e to a high numbrt of digits so let's focus on that. James has already nailed the method- you need to write your own routines to do the math. You need to implement add, multiply and divide functions that can work on numbers represented as strings in order to do the calculations for the taylor series he mentioned.
Build it up. Start with an add function- think about how you add two numbers, digit by digit and carrying the overflows, and come up with an algorithm that copies it. You'll need a loop to iterate over the digits and a shift register to pass the carries. Just have a crack at the add function. If you can do that much, you're well on the way to cracking the whole thing.
