"I am angry to die" or "I am angry to death"

Solution 1:

Your first choice of "I am angry to die" does not convey what you want to mean. It means that you are angry that you are dying.

One idiomatic way is to use the construction

I am so X I could Y.

X is an adjective (or adjective phrase) and Y is a verb phrase.

For example

I'm so hungry I could eat a horse.

Where X is hungry and Y is eat a horse.

So for your example that would be

I am so angry I could die.

Your second choice, "I am angry to death", is OK, but I don't think most people would use it.

Solution 2:

Americans like my mother say things like I'm sick to death of your moaning! I've never heard I'm angry to death. It's understandable but not idiomatic. People don't die of anger unless they have a stroke or a heart attack because of it (Type A personality people sometimes do). So I'd suggest you not use it.

I'm angry to die means that you're unhappy about the fact that you're about to die, so unhappy, in fact, that you're angry about it and want to curse God and whoever or whatever else might have caused your probably premature death. I don't think it's the right expression in this context. That should probably be I'm angry about dying (too soon / so soon / so young / for nothing / because I caught a cold / etc.).

Heartbreak Hotel (Elvis Presley)

Well, since my baby left me
I found a new place to dwell
It's down at the end of lonely street
At heartbreak hotel

You make me so lonely, baby
I get so lonely, I get so lonely, I could die