Since I got quite a bit of advice here, I'm doing a trip review.
I spent last week in Leipzig, travelling by plane to Berlin and then by train. I stayed one night in Berlin at the Motel One Hauptbahnhof. I wouldn't recommend it unless you're leaving by train in the morning like me, as the area is still quite undeveloped. The tram along Invalidenstrasse is now running, though, making transport to the rest of Berlin better. I did like the hotel itself and would try other branches of the chain. I had a good dinner at a place called Staendige Vertretung, on the bank of the Spree near Friedrichstrasse station - flammekuche with black pudding, apple and onion.
In Leipzig I stayed at the Days Inn, which is in a quiet area just east of the city centre, fifteen minutes walk or two tram stops from the Ring. The hotel is in a slightly modernised ex-DDR tower block, but perfectly comfortable and good value. The receptionists were very helpful - one warning me against travelling to the city my first evening due to a PEGIDA demonstration. It's close to the Grassi Museums, which cover anthropology, musical instruments, and decorative arts. I only visited the last of those, which has a particularly good art nouveau and art deco collection.
I do want to recommend a restaurant in the area which I thought was very good. It's called Zunftkeller, in the basement of the Haus des Handwerks. The food is local and seasonal, which in September translated to lots of game meat and wild mushrooms.
In Leipzig itself I also visited the general art museum, which I enjoyed, and the Runde Ecke Museum, which covers the nature and activities of the East German Stasi, or secret police. The labelling and the documents on display are all in German (and often in very difficult legalistic and political German) but you can hire a very informative English audioguide. It's in the former city headquarters of the Stasi, largely preserved and with a reconstruction of a detention cell from another building.
Thanks to recommendations here I visited Halle and the gardens at Woerlitz. Unfortunately the railway line from Dessau to Woerlitz is currently replaced by buses due to the railcars having broken down. But the gardens are still spectacular. I did a circuit of the inner parts but couldn't wander as far as I wanted due to suffering from backache.
Halle has some interesting older buildings surviving. I visited the Moritzburg art museum but at the time most of the permanent displays were replaced by an exhibition of Anthroposophist art, which didn't interest me a lot. I enjoyed the Bruecke paintings that were on show, however.
I planned to go to Zwickau for the Audi Museum, but discovered online that a large part of it was closed for renovation, so decided to leave it for a later trip.
Travelling back from Leipzig to London, I was very unhappy with the situation at Schoenefeld Airport. They are trying to push too many budget flights through what is still a very small ex-DDR terminal, and the queues for security and for passport checks at the gates were quite appalling.