My wife says that she divorced me by saying that saying “we are done” and that we have not spoken since then. It's been 2 months. I have been actively trying to reconcile.

She says that she does not need to ask for khula because according to her, I did not take care of her or do my responsibilities as a husband. She said that she does not need an Imam because she believes she has solid grounds to do so.

Is this valid? I know there is a method prescribed by Allah. What is the correct method?

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This is not the correct method of divorce. She is still your wife.

Divorce is the hands of the husband, unless he gives her power of attorney to perform the divorce, or a jurist conducts the divorce if the husband is not fulfilling his duties. However the jurist must reach out to the husband before conducting the divorce.

From what you say it seems your wife has not done any of those and therefore the divorce is not valid.