What is wrong with the following sentence?
Being unable to type the letter, the boss fired me.
1. It is a run-on
2. It is a dangler
3. It is a comma splice
4. It lacks parallel parts

It's a "dangler" or dangling participle.
Following "Being unable to type the letter" must come the main part of
the sentence with the person who was unable as the subject.
"Being unable to type the letter, I was fired by my boss."

That is not very good English, but correct.  I would choose to write: 
Since (or As or Because) I was not able to type the letter, my boss
fired me.