“I got mixed up and called my landlord by mistake.”

Earlier today, you meant to call one person, but you called someone else instead. Now you're telling your friend about it. You say this while telling your friend about the error.

I got mixed up and called my landlord by mistake.

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(someone) got mixed up and (did something)

To "get mixed up" means to confuse two or more ideas in your mind. Usually when you "get mixed up" it means that there are choices that you make a mistake and accidentally switch. For example,

I got mixed up and took Highway 50 South instead of North.

You tell what you did when you were confused with the phrase "and (did something)":

They must have gotten mixed up and tried to use Version 2 instead of the newer version.

call (someone)

You "call someone" on a telephone. You "call to someone" by shouting out loud from far away.

(do something) by mistake

When you do something that you didn't mean to do, then you can say that you "did it by mistake".