There are a number of issues in the suggested sentences unrelated to the point the question asks about.
In sentence 1, the normal uage is "on vacation" not "in vacation". Also, "two" should be spelled out.
In sentence two, a person would not write "The world war II..." but rather just "World war II..." or "The second world war..."
In sentence 3 the form should be "he drove" not "he drived". In fact I can't think of any context in which "he drived" is correct, at least not in US English. (Perhaps it is in other varieties.) My spell checker denies that "drived" is a correct form, but spell checkers never capture all possible correct forms.
In sentence 4 "they closed the hospital" should probably be "they had closed the hospital" as the closure is in the past.
Returning to the use of "which"
I lived in France for two months during which time I was on vacation.
The phrase "during which time" can be elided to "during which", because "two months" already tells us that the referent is is a span of time. The word "which" refers back to "two months". But the word "which" cannot be omitted, as it leaves the meanign of "during" unclear, and the grammar simply incorrect.
World war II, during which many people were killed, ended in 1945
"which" refers back to "World war II". It is better not to write "during which time" because the meaning is that people were killed in the war, not just during the time that the war was in progress. We aren't talking about people far from the front who died of old age or sickness in 1943. That would be "during the war", but not "in the war". All of which leads to a better option:
World war II, in which many people were killed, ended in 1945.
This puts the emphasis on people being killed in the war, which I believe to be the intended meaning.
He drove the motorbike at a speed which no one had reached before.
"which" here refers back to "speed". Many people might phrase this as "...at a speed that no one..." However, "that" should properly only be used with a restrictive clause in such a construction. This rule is being ignored more and more, and may eventually become obsolete, but it has not done so yet.
One wold not say o5r write "at which speed" here, because there would be nothing for "which" to refer back to. It could be used in a different construction, such as:
He sped up to 130 MPH, at which speed he lost control of the car.
In that example "which speed" refers back to "130 MPH". Or if you like a more positive example:
The rocket accelerated to more than seven miles per second, at which speed it was free of the Earth's gravity.
Here "which speed" refers back to "seven miles per second".
Finally I come to sentence number four. Here the intended meaning is not clear. Did she call the doctor before the hospital was closed, or after it was already closed? If before, one might write:
She had already called the doctor by the time they closed the hospital.
or if you want a form using "which":
She called the doctor before the time at which the hospital closed.
But the first form is more natural in my opinion.
If she did not call until after the hospital closed, one might write:
She called the doctor after 10pm, by which time they closed had the hospital.
Here "which time" refers back to "10 pm". If the time of the call is not to be specified, one might write:
She did not call the doctor by the time the hospital closed
She called the doctor at a time at which the hospital had already closed.
This is a forces construction to bring in "which" and would be more naturally written or said as:
She called the doctor when the hospital had already closed.