Sturmey-Archer three-speed to Single-Speed

Hmm, without the indicator chain it should just run in high gear... shouldn't it? Perhaps the hub is all gummed up inside.

Anyway, the chains are cheap and available on ebay as oller says.

The advised setup is that the cable be completely slack in top gear which would suggest that loose indicator chain/no indicator chain should yes, run in top gear.

I'd suspect another problem with the hub.