「……さらわれて？兄様、どういうことでいやがりますか？」
「ーーミオは……どこにいるんだ？あいつが助けてくれたんじゃないのか？」
「だから、何を言ってやがるんですか、兄さーー」
I know 「どういうこと」 means “What do you mean” and 「やがる」is her 口癖. But I cannot understand the 「でい」.
Maybe it’s「どういうこと」+「でいる」+「やがる」? And this 「でいる」 is like "to stay/keep ～", as in this thread? https://japanese.stackexchange.com/questions/66270/what-is-the-difference-between-でいる-and-である-in-this-example
やがる describes something while expressing disdain/contempt/hostility. For example, if you say 休んでいやがる instead of just 休んでいる, it means you consider them to be enemies (or at least subject of your contempt), and implies you'd rather see them not 休む.
Here, this やがる is shoehorned onto どういうことだ. This sounds odd because it should be hard to feel contempt towards something you don't even know. It's even odder because it's combined with a question (どういうことでやがりますか？). Usually if you are using やがる your main point is expressing contempt (rather than ask a question at the same time).
This is done here because the author is using this form to add a distinct speech to the character (so the character adds it no matter what the context is because it's their "signature speech").
Coming back to your specific question, the "normal form" would be どういうことでありますか, but it's very wordy to say どういうことでありやがりますか, so the author opted for どういうことで居やがりますか. IMO どういうことでやがりますか would have been more natural, but whichever form you use it's unnatural anyways. I think the author just liked the former form better. You won't hear this expression in real-life.
