You should try. Washington's Pattern Jury Instructions allow for the following:
Reasonable compensation for any loss of use of any damaged property during the time reasonably required for its [repair] [replacement]. (WPI 30.16)
Our Supreme Court said:
We now hold that, where, as here, a plaintiff has not rented a substitute automobile, he is nevertheless entitled to receive, as general damages in the event liability is established, such sum as will compensate him for his inconvenience. Proof of what it reasonably would have cost to hire a substitute automobile is sufficient evidence to carry this item of damages to the jury, but is not the measure of such damages. It is relevant evidence in determining the general damages for inconvenience resulting from loss of use of an automobile.
Holmes v. Raffo 60 Wn.2d 421, 431, 374 P.2d 536 (1962).
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Like everything else, it is a negotiable item of damage. It is called "loss of use", and you should document your claim by obtaining the average cost of a rental from several local rental facilities. This will give you a reasonable basis to discuss your claim. If you fail in your negotiations, make it a part of your bodily injury damage claim. Sooner or later you will likely receive some amount whicy will be satisfactory. Good luck!
If your wife incurred additional auto expense in ferrying you to and from work, etc, then this would arguably be a recoverable incidental expense, even though not technically a rental car expense. Sometimes it's a matter of how you phrase the request.