bad, badly

Bad or bad describes something -- something of unacceptable quality, something spoiled, something unpleasant, something or someone disobedient or wicked --
This is such a bad movie!
I think this milk is bad.
He is having a bad day.
What is that bad smell?
He made a bad decision.
We should correct the dog's bad behavior.
Bad dog!

Badly or badly describes an action -- an action done poorly, ineptly, unsuccessfully, or incorrectly --
He did the job badly.
He plays tennis badly.
We planned this badly.
He behaved badly.

Now that you know that, you can say--
"He feels bad that it turned out so badly."

Note -- It can be difficult to decide whether a word is an action or a thing. Feel is one example of this. Most people agree that when feel means an emotion, it is described as a thing --
"I feel bad about the pile of work I left on my desk."
When feel means to touch something it may be described as an action --
"He thought this was real fur; he feels badly."

Note # 2 --"badly" also means seriously or gravely --
"The building was badly damaged in the fire."
"He was badly hurt; he will be in the hospital for weeks."