Can you start a sentence with "which" when not asking a question?
Which way you answer that question depends on the perspective you bring.
Won't that work?
Or, 'The colour of tie you choose will probably depend on your mood.'
The OP used "which when it's not a question word" - and so didn't exclude the acceptable use when "which" is [implicitly] a question word in indirect speech.
(Of course, this only applies to the written language. If I write "She came too - which was lucky.", that's grammatical. But "She came too. Which was lucky" wouldn't be OK in a formal context. The words are the same though, and the meaning's the same.)