I so enjoy the little challenges that you put here and which test the knowledge of natives and non-natives alike. No comma and no ‘that’, but an ‘and’ and a ‘which’. What now?
Ashenden: Or the British Agent
SANATORIUM
http://www.aaoldbooks.com/en-uk-us/Creatures_Of_Circumstance/book_p_086.asp
“For the first six weeks that Ashenden was at the sanatorium he stayed in bed.”
For the first six weeks–Ashenden was at the sanatorium, he stayed in bed.
The clause “that Ashenden was at the sanatorium” is, as a restrictive relative clause, modifying or restricting “the first six weeks,” which is not a clause, and there’s a preposition “for” before this. If you think that “that” as a relative adverb cannot be replaced by “when,” I wonder whether you might suppose that you cannot begin a restrictive relative clause with “when.” Or, “when” following “for” might simply be considered strange.
In your example sentence, the first version uses “which” incorrectly, making it sound terribly awkward, and as a native speaker I would never say that. “She held out the hand that was hurt” is the correct one.
@brucenator
At last I was able to understand what you wanted to say. You think that the clause restricts or modifies the six-week period and that you cannot begin a restrictive relative clause with “when.” I wonder whether you think that if you begin a relative clause with “when,” it does not restrict the period but gives additional independent information, and that Ashenden will be supposed to have stayed at the sanatorium for six weeks, that is, only the first six weeks of his longer journey or something.
I think that you used “that” and “which” to begin restrictive relative clauses. I suppose you did not want to use “that” twice, so you used “which.”
“At last the day came that he had been patiently looking forward to when the doctor told him he could get up; . . .”
In the above sentence, both “that” and “which” are used to begin a restrictive relative clause modifying “day.” I suppose the writer did not want to use “that” twice, but I am not sure.
P.S.
In the above sentence, “that” is used as a relative pronoun.
There is no “which” in that sentence. You’re confusing yourself now! 
It isn’t just a matter of preference in this case. Using “that” again would be incorrect and awkward sounding.
“That” refers to the specific day, “that day” he was looking forward to.
“When” refers to the specific moment in time that day “when” the doctor told him he could get up.
" ‘When’ refers to the specific moment in time that day ‘when’ the doctor told him he could get up."
I cannot agree with you. I think that the when-clause refers to, or modifies “day.” By the way, I am not confusing “when” with “which,” which is a relative pronoun, not a relative adverb.
“At last the day came . . . when the doctor told him he could get up; . . .”
There is an archaic use of ‘which’ which can serve to illustrate the use of ‘the day … when’: “At last the day came on which the doctor told him he could get up”. This way it is clear that ‘when’ refers to the long-awaited day.
P.S. I used the double which as a bit of fun.
Edited for double ‘day’ 
I agree that “when” begins a clause modifying “day.”
Replacing “when” with “that” would sound wrong, even if the first “that” clause were absent.
“At last the day came when the doctor told him he could get up.” is fine.
“At last the day came that the doctor told him he could get up.” is awkward.
“’ At last the day came that the doctor told him he could get up.’ is awkward.”
I think that if you omit “that,” it will become more awkward.
“At last the day came the doctor told him he could get up.”
Do you think that this is the same awkwardness that you feel when you read the sentence “At last the day came that the doctor told him he could get up”?
I think it is less awkward to omit “that”, but still not right, because “that” shouldn’t have been used in the first place. It requires “when”.
To clarify my previous posts, “when” describes the day by introducing an event that takes place during that day. It’s referring to a period of time (that day), so “when” is used.
“The day came that he had been looking forward to” describes the day, but not an event that takes place during that day, so “that” is used.
In the case where “that” should be used and directly follows the subject, it can be omitted. “The day he had been looking forward to…”.
But “that” is used here to connect back to “the day”; “At last the day came that he had been looking forward to.”
If “when” should be used, on the other hand, omitting it will leave the sentence incorrect. “At last the day came when the doctor told him he could get up.” Here, you can’t omit “when” or it would become a run-on sentence.
They’re both wrong. I don’t think debating degrees of wrongness is helpful.
If you don’t want to include any conjunction at all, make it two sentences:
“At last the day came. The doctor told him he could get up.” Which would be fine.
"To clarify my previous posts, ‘when’ describes the day by introducing an event that takes place during that day. It’s referring to a period of time (that day), so ‘when’ is used. "
“When” does not necessarily introduce “events” that take place during a period of time.
“He could remember a time when he had worked like that himself.” —Collins COBUILD ADVANCED DICTIONARY
P.S.
I will add the following:
“It was a time when people didn’t have to lock their doors.”
“the happy days when we were together”
Your example sentence seems to do precisely that. He “worked like that” (event) at “a time” (period).
The time is not specified like a day, month, or year, but it refers to a certain period “when” he “worked like that”, which is the event, although continuous.
“the happy days when we were together”
When-clause does not necessarily introduce events. Being together cannot be considered an event. (In addition, that-clause can also introduce events.)
- “the happy days when we were together”
- “the happy days that we were together”
- “the happy days we were together”
Could you tell me which is correct?
"They’re both wrong. I don’t think debating degrees of wrongness is helpful. "
I do not intend to debate degrees of wrongness. I tried to know the common element that is related to the awkwardness of some expressions in English.
Unless you ask a grammarian, which I am not, at some point it just comes down to “It sounds wrong.”
How do I know it sounds wrong? Because I’ve been listening to spoken English my entire life, and people don’t talk like that.
Once you work your way through Maugham, you might try Hemingway, then Faulkner, just to see how vast a range of styles English allows. (Start with Faulkner’s stories and novellas, though. A lot of the novels are impenetrable even to native speakers.)
"Unless you ask a grammarian, which I am not, at some point it just comes down to It sounds wrong.’ "
I agree. I don’t want to argue about things related to Japanese grammar, which I almost forget.
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