This is a very rare word from an old book. It means "a little stream which feeds a water-mill". I've never seen it before and I doubt I'll ever see it again.
(Also archaic: "the Thames could not peer with..." meaning the Thames could not compare with...
The Thames is a proper name, but the mill-streamlet is not. A mill-stream is a stream (small river) that was used to power a water mill. The suffix '-let' makes it smaller: a streamlet is a small stream. (This is not a common word.)