What is the best neighborhood to stay in if we only have 2 nights?
city centre (Śródmieście)
We stayed in the area south of Jeruzalemskie street on Nowy Swiat. From there we could walk the Royal Way (mostly pedestrian) all the way to the Old Town. This way passes through shopping, restaurants and major sights. Essentially that street has much of what you want to see, and you dont have much time to wander further.
As Stan suggests, staying close to the intersection of Jeruzalemskie and Nowy Swiat puts you close to major sites, many choices for dining, and excellent transportation options to/from the train station and up Nowy Swiat to the old town. It's also relatively easy to get to the Uprising Museum using tram or bus.
Thank you. I'm finding many excellent and pretty affordable choices in this area. Looks convenient to everything.