To find who owns the land, use the facility provided by the Land Registry.
The starting point is here: -
https://www.landregis...ortal/Property_Search
You will probably have to submit a Map Enquiry (because there is no detailed address). Quote it as 'land to the south-west of [your property address]. The LR then search their records.
If they find a proprietor they will tell you the registered address of the owner and you can write a letter expressing interest, as you proposed. Don't talk about prices until you have got them thinking about selling. There is no formula for establishing a value.
If LR don't find a proprietor, one option is just to fence the parcel of land and start maintaining it as your own and see what happens. The 'owner' may show up, or he may not. Then come back in 10 years and w'll advise you what legally to do next.