The National Cycle Museum is the strongest reason to stop in Llandrindod Wells. Its collection runs from early hobby-horse machines and penny-farthings to modern racing bicycles, with enough odd frames, accessories, and period detail to interest people who do not normally care about bicycles.
The setting helps. The museum sits in the former Automobile Palace, a showy early motor-and-cycle showroom that fits Llandrindod's spa-era confidence. Allow more time than you expect if you like design history or old engineering.