I was married for 19 years, during which time my ex-husband was playing cards every Friday

This relative pronoun could easily be replaced with ‘this’ or ‘that.’

Why does ‘which’ make it a dependent clause but this or that would make it a complete sentence?

Thanks.

I was married for 19 years, during which time my ex-husband was playing cards every Friday

I think the following explanation is right. (It’s the only one that I can think of.)

The two root clauses in your sentence are

I was married for 19 years
and
My ex-husband was playing cards every Friday during the time [that] I was married

Change [that] I was married to to which, and it becomes during which time.

So what you have is a long, dependent noun clause serving as the object of the preposition during. The result is a complex sentence made up of a main clause + a dependent clause, which requires separation by a comma.

Changing which to this or that makes to the two root clauses

I was married for 19 years
and
my ex-husband was playing cards every Friday during this [or that] time.

Here, you have a two main clauses requiring separation by a period.

Yeah, that was a hard question. Well done. It sounds right to me.

Cheers.

That was hard to figure out because, on the surface, the sentences appear so similar.