Wat Thammikarat is just outside of the eastern wall of the royal palace, near the river. It is thought that the temple may pre-date the founding of the new Siamese capital at Ayutthaya, but the temple was incorporated into the new city.
Wat Thammikarat appears to be largely in ruins, but it is in fact an active temple with monks in residence. Several new structures have been built although very little has been done to restore the ruined buildings (temples can't restore historic buildings without the government Fine Arts Department, which doesn't have the resources).
The eye-catching part of the temple are the ruins of the giant prayer hall (wiharn). Massive columns line the sides of the building while two rows of even larger columns march down the interior of the hall. The only windows in the side walls are narrow slits. The atmosphere of decay is further enhanced by the presence of several trees growing out of the platform near the rear wall of the wiharn.
There's no cost to visit the temple, but since it is part of an active monastery you should show the appropriate respect.