Weybridge is currently beyond the contactless area, so you’d need to buy a ticket for at least part of the journey. I think the cheapest reasonable route is likely to be via Wimbledon. You’d need an Anytime return from Weybridge to Wimbledon which is £12.60, then use PAYG from Wimbledon to South Kensington on the District Line which is £3.70 peak (£3.00 off-peak if you touch in after 1900 at South Kensington). Total £20.00 (or £19.30) which is quite a bit cheaper than the Anytime day travelcard at £29.40.
If your commute is generally 5 days a week then a season ticket for Weybridge to Wimbledon will save a bit, but I’d still use PAYG for the Underground.