Have a trip next Summer for total of 17 nights, options for packing and doing laundry

You’re likely going to need to pack light for your 17 days. One generally can pack 5 underwears or 2 Ex-Officio pairs. (They’re nylon and easy to sink-wash& dry) 4 pairs of socks. (Nylon if you can ‘cuz cotton socks are bulky) and 4 shirts and do laundry on days 5 and 9. Two pairs of walking shorts and you’re good to go.
Pack a 3-ounce plastic bottle with liquid laundry detergent.with Oxiclean. On a sunny day ( they do occasionally have them in the UK and Ireland) it’s time to wash and dry. Otherwise, it’s off to a laundromat or cleaners.
The UK and Ireland can experience heatwaves, but usually you’re not going to be sweating in 90-degree weather as you would be in Italy or Spain during August.

Wear your bulkiest pants, thin merino wool sweater and windbreaker jacket to the airport and shed layers before the flight leaves.

Pack any excessively-heavy items in your “Personal Item” which airlines never weigh.

And—although it will be summer—-leave your white clothes at home!

Are any of these near your hotel?
Laundrette
6 Lupus St, Pimlico, London SW1V 3DY, United Kingdom
Pimlico Laundrette
3 Westmoreland Terrace, Pimlico, London SW1V 4AG, United Kingdom
Launderette - Westminster
13 Regency Pl, London SW1P 2EB, United Kingdom

We have found no success in sink washing and having things dry out in small bathrooms in hotels. You will get lots of experienced travelers who handwash, but we find it unsuccessful, and frankly I just don't feel like I'm wearing clean clothes when I do that.
I would call/email your hotel in London and ask where the closest laundromat is; or find out if they send laundry out. Many hotels offer this service.

I pack a laundry bag forr dirty laundry and a clean bag to carry the clean clothes back from the laundomat.

...but will they dry in a day and a half?

You know those towel heater/dryer radiators in UK hotel bathrooms? I've used those to dry articles that I've hand washed.

Yes, ask your hotel if they have laundry service. In Amsterdam I asked at the registration desk about nearby laundromats and was told they had someone who would wash and return what was a good-sized bag of laundry for 20 Euros. It was well worth it - not least of which because I had never used a laundromat in my life and was a bit intimidated by it.

I used laundryheap earlier this month. They will pick up and drop off at your hotel’s reception desk.

There’s an app for the phone or you can probably set it up from the website.

You can enter your hotel name to see if they will pick up there.

https://app.laundryheap.com/booking?city=1&_gl=1%2a4dy21q%2a_gcl_au%2aMjAyMDI3MzMwMS4xNzI5MjkzMTY0%2a_ga%2aMTA1NTgzODYzMi4xNzI5MjkzMTY2%2a_ga_5099ZG1JDG%2aMTczMDMyNjU1Mi4yLjAuMTczMDMyNjU1Mi4wLjAuMA..#step1

It worked great and was super convenient.

Is that 2 days and 1 night you’ll be in London? Or 3 nights and 2 full days? If it’s just 1 night that is probably not enough time to use the service. It’s a 24-hour service, but we didn’t have a choice on time. I think our pick-up time from the hotel was 3pm, which meant it would not be delivered until 3pm the next day.

Assuming you're flying in and out of Heathrow, you could store a bag (full of dirty clothes from Ireland) at the airport (or at another location in London that would be convenient to pick it up after the Rabbie's tour) for about £100 or so. I know some folks on the forum ship their dirty clothes home. I can only imagine that shipping a suitcase home would cost more than storing it for a week.
Or you can figure out how to pack light. This site and the wider web are FULL of information about how to pack light.
If you hand wash your clothes and they don't dry in a day and half (which you can test at home, BTW), you're probably not packing light enough.

Some ideas for your wife: If she likes to wear easy-going dresses in the summer, this is my “go to” dress. I bought it in navy a year ago and have worn it in England, Wales, Slovenia, Croatia & Italy. It is dry after sink-washing in about three hours. I pair it with some bouncy earrings or a cute scarf for variety. I literally could wear it every day on warm trips! I traveled twice in the last year for a month at a time with just this dress, a black lightweight dress, and two pairs of pants…and I always sink-wash my dresses, underwear and shirts after each wearing.

Royal Robbins® Spotless Traveler Tank Dress – Scoop Neckline – Snap-Button Placket – Blended Polyester

For you: my husband liked to wear very lightweight golf pants on trips. They would dry overnight when he sink-washed them. For shirts, he liked the closer-fitting Underarmour type T’s or golf shirts.

Usually weighing each item with a kitchen scale will give a good clue how quickly an item will dry. I weigh all of my possible options I like and select the few that weigh the least.

The simplest solution is just to take a taxi to the nearest launderette that will do a wash and dry for you. Drop it off, then pick it up a few hours later. Or plan to spend a couple of hours there to do it yourself. I see 4 launderettes near Victoria Station. We did a launderette run in Edinburgh after a 10 day tour, and were able to get all our shirts and pants into a single load. I always sink wash my own lingerie and socks, and have always had them dry by the next day. DH will sink wash his underwear and socks, and finds they take about half a day longer to air dry. If you have a heated towel rack in the bathroom, use it for any items that are slower to dry.

My husband has a couple of Royal Robbins shirts (short-sleeve, buttonu-p) and they are great for traveling. Easy wash, quick dry, non wrinkle, and look really nice.

Tech fabrics are your friends.

My luggage for 4-1/2 months in Europe was one carry-on; my wife's was one carry-on plus a backpack. We did have washing machines most places we stayed, but we could've washed in the sinks if we had to; I did so on a month-long trip to Spain a year ago with the same luggage.

My underwear for travel is all nylon Body Glove; my shirts are golf shirts of tech fabrics; my trousers are Royal Robbins tech fabric. Socks are similar. My wife's clothes are of comparable materials. They all dry completely in a few hours if they're twisted dry using hotel towels; you may need to get extras from the front desk. I also carry a microfiber travel towel from Mountain Warehouse; other outdoor supply stores have similar products. Use shampoo for washing, as its detergent qualities are excellent, it's gentle on the clothes and it usually has a nice aroma.

As for shoes, I wore one pair of running shoes and packed slippers in my bag for indoor use. No one needs more shoes than that. Trying to "not look like a tourist" is a losing battle under any circumstances. I wore the sneakers to the opera and two chamber concerts, and no one cared.

I dont usually get into the clothing discussion, but what the heck, I am waiting on a delivery of lumber from OBI and nothing to do.

Summer is easiest for me to pack.

I most often plan a nice evening or two so I have a light wrinkle resistant blue sports coat (that I wear on the plane - or sometimes I pack) and a pair of white no-iron dress trousers and a pair of classy woven leather sort of like dress sandals for a guy; then a slick looking "salmon" (guys dont wear pink) crew neck shirt. And I look all Miami Vice at dinner. If its the opera the trousers are grey the shirt becomes a no-iron white dress shirt, and the shoes are usually a pair of Lucchese's (that I wear on the plane).

Then there is one pair of jeans and the Stetson beaver (that I wear on the plane)

Then everything else is from Bass Pro. Fast dry nylon fishing attire. Wash it by 10 pm and dry by 6 am. Just difficult to find the shirts without the big vents in the back. Dont like those at all but Columbia makes one line that is almost dress shirt looking (almost).

8kg including the 2.6kg weight of the bag plus about 4kg in my personal item. Three weeks is no problem. I am guessing you wont be taking the Stetson or the Lucchese's so ought to be easier. 5 fishing trousers = 1 pair of jeans in weight.

That was fun. Thank you.

You could check with one of the hotels you're staying in to see if they have or can arrange laundry service and what the turnaround time is. When I used to cruise with my mom, we would get the laundry done. We put our "unmentionables" in a zippered mesh bag.

It's been a few years (2019, I think), but we used a pick up/drop off service in London:
https://theempressofsmoothprofessionalironing.freeindex.co.uk/
They picked up during the night at our hotel near Covent Garden (we left it with the front desk) and dropped off the next evening, iirc. It wasn't cheap, I think about 50 pounds, but it was 10 days or so of laundry.

As others have said, invest in some quick dry fabrics (I prefer cotton in everyday life but choose mostly quick dry for travels) for shirts, dresses, pants, skirts & underwear. (I still need my wool, partially wool, or mainly cotton socks.). Take a couple of "plastic folding travel hangers" from Amazon; roll in towels, use towel warmers, and a blow dryer to speed the process if necessary. The right fabric is easily washed and should dry overnight.

For a 2nd pair of shoes, I often take my keen sandals (maybe Keen Whispers) so I can go without socks, rain or shine. They don't take up much room.