Learn English – What’s the meaning of “floor discussion”

The second usage of "just" is being used to minimize the request as a whole. In other words, it is saying "what I am asking for is really a small thing and it costs you nothing". It suggest that what the person is asking for is trivial or harmless or costs nothing and therefore there is no reason not to grant the request. Thus, Cyndi Lauper sang "Girls Just Wanna Have Fun".

So in this case it is modifying "use", but not in the sense of "do this thing and no other things" but in the sense of "do this trivial thing that costs you nothing".

"By which time" refers to at/before a particular time that has previously been mentioned. In the sentence, it's the time she finally called the solicitor.