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はい、こちらサービスセンターです。
すみません。コンピューターが壊れてしまったんです。
どこが壊れたんですか。
わかりません。
As far as I understand, the manager is asking what exactly has broken (which part of the computer is broken?). If we said 何が壊れたんですか in this situation, would the answer be the computer?
When shall we use どこ and 何 in subject questions?
Yes, your assumptions are correct.
As 何 just means 'what', asking 'What is broken?' would be met with incredulity, as you already told the rep that the computer is broken.
（～の）どこ in cases such as this these, means 'which part of'. In essence, part of the subject is being omitted in the query from the rep. The full sentence would be 「コンピューターのどこが壊れたんですか。」
This use of （～の）どこ is fairly ubiquitous for specifying the part or specific location of something ('Where does it hurt?', 'Which section/area of the park should we meet?', 'What don't you like about me?', etc.)
In this context it would be odd to ask 何が壊れたんですか？ because the answer would obviously be コンピューターです
Customer: My computer is broken.（コンピューターが壊れてしまったんです）
Service Tech: What's broken?（何が壊れたんですか？）
Customer: I just said it's my computer!（コンピューターだと言ったでしょ？）
A better response would be...
Service Tech: How is it broken? (What's wrong with it?)
The way to ask this in Japanese is to use どこが壊れたんですか？
You should use どこが when you want to know "what part of" or "in what way" some state came to be.
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