6 nights in London- do we need a bigger place?

Did you book the Royal Mews Double, or the Royal Mews King? Quite a difference in the size of the room.

Dimensions of King: Room size range: 17.5m² / 188 ft²
Bed size: 180cm x 200cm

The double room seems quite small at 135 Sq ft.

Only you can decide whether those dimensions are acceptable. Plot the hotel on a map to determine whether it's close enough to a Tube stop to suit the demands of your trip.

In central London you are never far from a tube station and some action.

My experience in London is that after a long day of sight seeing, we enjoy going back to our room, drop the shoes, make a cuppa (Tea or Coffee) and relax before going out in the evening. A pair of arm chairs and a kettle are essential for us. But we are about 20 years older than you. YMMV

The Rubens is a lovely hotel. I had a last minute/emergency stay there a few years back and they were so helpful and welcoming.
Some of the rooms are quite cosy - consider a superior king if it's in your budget.

that double converts to 12.5 m square

That's pretty compact. I always go for at least 18, especially for a few days

I don't know you and your travel style, but if you think you might want to hang out in your room and sit on something other than the bed, you'll need a room that has somewhere to sit besides the bed. They do have a larger king room with a "sitting area," but even that looks pretty small. I'm assuming you picked a luxury hotel because you're celebrating this big birthday. If it were me, I'd splurge on a celebratory meal (and other experiences) and stay in a place where I felt comfortable in my sightseeing clothes at breakfast, in the bar, etc., and where I could comfortably hang out in my room with a bottle of wine and some snacks from Tesco. That's my travel style.

@G3rryCee thank you! your description totally fits the way we normally travel- I think I just got pulled into the Rubens having a sale and being so nice to us and then it being London and wanting to stay somewhere "Londonish" haha if you have a place to reccomend that matches what you described I am all ears!

I don't have a hotel to recommend. You can do a search on booking or similar site and filter for king bed, price, etc. From there, when you click on an option, it will show you the room dimensions. I did this for 6 nights in October (not your dates, probably), and the first one I saw (Apex Temple Court) had a room size of 269sqft, which is bigger than the large room at Rubens. The same hotel had a 400sqft suite for $445/night, which is still below your stated budget. I only use booking to find hotels. I prefer to book directly with the hotel if it's a private one like Rubens. If it's a chain, I don't mind using booking (or lately, Trivago) to get the best price. Wherever you decide to stay, have a great trip! And happy birthday to your husband!

I think 6 nights in a small hotel room is a long time- I would feel cramped.

Is a hotel situation a “must have” for you? I would consider a flat for a 6 night stay. You could have a nice flat and still come in under your generous budget.