For the proposed visit, you will need a Standard Visitor visa.
While it is possible for people holding a US visa to request leave to enter the UK to transit the UK landside, without holding a visa, V 7.5(a) of the immigration rules requires that in this circumstance, the reason for entering the UK is to transit.
V 7.5 An applicant must satisfy the decision maker that they:
(a) are genuinely in transit to another country outside the common travel area, meaning the main purpose of their visit is to transit the UK and that the applicant is taking a reasonable transit route; and
Because you are not changing airports in the UK, nor do you need to collect luggage landside and check in to the next flight as on separate tickets, it's not necessary to leave the transit area of the airport in order for you to complete the transit. The main purpose of leaving the airport -- as you have described -- is sightseeing. Thus, you will need the Standard Visitor visa.