Advice Struggling to Book a Room for 3 and Train to Paris

Greetings, we'll be in London from late December into January and I'm having trouble finding a room that accommodates 3 people. Also, I need to book a train to Paris during that time. Does anyone have recommendations for family-friendly hotels in London that can comfortably fit three, and any tips for booking trains to Paris around the New Year? It seems like everything is either fully booked or super expensive. Any advice or alternative options would be greatly appreciated!