Is it DIYable? Yes. No. Maybe?
You may find that you start the job, everything works out just right, and you complete the work in record time and under budget. More likely, you'll run into a bunch of unforeseen problems, take 5 times as long as you'd expect, and end up way over budget.
The best way to tackle DIY work that you are not sure about, is to have the resources available to quickly fix mistakes. In your situation, it might be a good idea to purchase a replacement window before you begin. If you damage the old window beyond repair, you'll simply have to install the new one. On the other hand, if the repair works out, you can return the new window you purchased.
It also never hurts to have somebody with knowledge in the type of work you're doing, so if you run into trouble they can offer guidance.
If you come across something you are really not comfortable with, there are always professionals available to do the work for you. You might also be surprised that they charge less than you might think (not always the case, but does happen).
I like the answer JonW gave, but also would suggest just putting an awning over your existing window and opening it from the top as it's likely to be a bit less expensive.
Edit:
I passed this house fixed type awning system and decided to grab a shot to add t this post
Best Answer
Are you sure that the weep holes are not blocked? This issue should be notated in your owner's manual. Otherwise, the window may not have been correctly sloped and flashed in the frame, and it took this long for water infiltration to evidence. Is there rot on the underside of the window, when opened? Older Pella awning windows had a problem similar to this.