Traveling to London with autistic child

I’ve got an autistic child who needs some quiet indoor space to walk around in private. He doesn’t make any noise, he just likes to walk in circles without anyone looking at him. I’m searching for hotel rooms that have space to walk around the furniture. Sadly public spaces just wont cut it, as it needs to feel like their personal safe area. Most people just use hotels to sleep in, but we need a bigger space. But we don’t need luxury! I’m hoping to find a place that’s $600 a night or cheaper that is roomy. Any ideas? We need to sleep 3 people (in beds, not pullout sofas), and we need to be less than a five minute walk to public transit. Am I looking for something that doesn’t exist?