Am I using "as best I can" correctly?

It's better if you say: "My colleague had a bit of an accident on his way to work and got himself injured. So I took out the first aid kit and patched him up as well as I could."

or as in your second sentence: "Mike really wanted to make her birthday special and he went all out for the party. He baked the cake, prepared the food and drinks, and decorated the room as well as he could"

Adding that the is a possibility of including "doing my best" instead of both terms.

To answer a question that may arise, as to why these two phrases may be better than the one you used, is due to the opinion that "as best I could" has become a redundant phrase, or in a better-described manner, it has become superfluous.