First, what type of High Deductible Health Plan did she (or you both) have? Self or Family? NOTE: this is the type of policy, not your filing status.
The annual HSA contribution limit for Self-only is $3,850; the annual HSA contribution limit for Family is $7,750. If your wife is 55+ then there is an increase of up to $1,000 a year in the limit.
She is covered by her HDHP policy for any month in which she was covered on the first day of the month. Thus, to be covered by August, she needed to be covered by August 1st, otherwise, her coverage started in September.
The HSA annual contribution limit is prorated by month. So take the appropriate annual coverage for Self-only or Family, then multiply by the number of months she was actually covered and divide by 12 - and this will give you her annual HSA contribution limit for 2023.
You should be able to reproduce this by subtracting her contributions from the limit you just calculated.
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