Learn Spanish (Latin America): Él puso mal el clavo, de manera que volvió a romper la silla. - He put the nail in wrong, so he broke the chair again.

The phrase de manera que can trigger either the indicative or the subjunctive depending on the context. We use the indicative (like the preterite volvió here) to talk about the actual result of an action. We use the subjunctive if we are talking about an intended purpose. Since he didn't put the nail in wrong on purpose just to break the chair, it's an unintended result, so we use the indicative.

Could I use así que instead of de manera que in this sentence?

Yes, así que also means so and works perfectly here to show a result. De manera que sounds slightly more formal or descriptive, literally translating closer to in such a way that, but they are completely interchangeable when talking about actual results.

Why doesn't the sentence say rompió otra vez for broke again?

You certainly could say rompió la silla otra vez or rompió de nuevo la silla. However, using the structure volver a plus an infinitive verb (here, volvió a romper) is a very natural and common way in Spanish to express doing an action again.

Why is it puso mal rather than using a word like equivocado?