I don’t think it’s a good idea to combine the airport shuttle with the rest of your journey. The fare for adults is £2.40 single or £3.80 for an open return (should be valid for a month). Children are half price.
For the rest of your journey I can’t actually see your quoted prices. I think the £37 price may be wrong. £12.60 is correct for contactless from Luton Airport Parkway to Waterloo using the Underground from Kings Cross. But there’s an even cheaper way. Stay on the Thameslink train to London Bridge and then pick up any Southeastern train to Waterloo East. This avoids the Underground and so saves you £2.40 making the fare £10.20. The main exit from Waterloo East is onto the concourse at Waterloo Mainline.
For your child it’s a bit more complicated. Contactless only charges adult fares so you’ll need to buy them a ticket. Unlike contactless you can’t avoid an Underground inclusive fare for Waterloo East, even though you don’t actually use it. The £10.30 price is from the Airport including the shuttle bus. Separately the bus is £1.20 and the train from Luton Airport Parkway is £9.05, so a 5p saving. If you get the return on the bus it’ll save a bit more over both journeys.
So to summarise, shuttle bus returns £3.80 + £1.90, contactless for you £10.20 and ticket for child £9.05.