Girlsdad, you sound like a good father and husband.
I agree that sleep time IS me time at this stage. Please advice your wife to have her lunch while baby is awake, it is OK to let baby watch some tv, or sit in a bouncer and watch mum eat. They learn from watching others, and seeing mum eat "normal" food is not a bad thing.
We had a "bumbo" seat which I kept baby in while I had a shower.
As for housework, and help, do you have a cleaner? Can you afford one just until baby is older and sleeps better? It might take the pressure of your wife a little. Also, our cleaner would also do some babysitting. When we got to know her and felt comfortable around her, she would look after baby for a little while I had a rest. I am not suggesting you pay a cleaner to be your nanny, but it worked for us, as we were all in the house, and it was only for an hour a week.
Do you help with the night feeds, or is it only your wife getting up? I know you need to feel alert for work, but she also needs to feel alert for her day, so is there any way you could share this a little? Like, you go to your dd the first time she wakes up?
Also, please advice your wife to go to bed early. Even before baby is in bed. Could you put baby to sleep before her last feed, before you go to bed?
You can also ask the health visitor to refer you to a sleep clinic, they are really helpful.
I breastfed mine, and he would wake up 3-4 times per night for feeds until he was 12 months old, I was exhausted! We got some brilliant advice from our gp which we followed, and our baby settled to sleep through the night within a week. 5 months is too you for this though.