Sounds like Prague would be a top choice! At that time of year, you should be able to find cheap-ish accomodation as well (once the huge craze over New Year has died down) and there may even be cheap flights there, have a look through the cheap airlines (wizz air maybe?) .
I would say that Prague ranks right up there as one of the prettiest cities I have been to. An amazing castle and old town, huge cathedral, the bridge. Lots of chances just to wander about and discover small side streets with their homely pubs. Czech beer leaves little to be desired and is reasonably priced, especially if you venture a little off the tourist trail when you're there. Lots of music events, concerts etc as well.
Once you're there, it's not too hard to get around other parts of Europe as well. Western Europe may seem a tad boring, but Munich or Vienna are in easy reach and well worth a trip. I'm sure I don't have to tell you about the amazing beer you can get in most of Bavaria, if you like Erdinger and it's siblings. Budapest is also said to be very interesting and is quite close, although I haven't been and I am not sure how easy it would be to get around there (as for languages, some German allegedly helps in large parts of Central Europe still!)
Or something completely different - consider Scandinavia! Cheap flights there, and travel between the cities is quite easy, but these are some of the most expensive countries in Europe, and alcohol and travel especially could easily blow your budget. Think ~5-6 pounds for a pint in Norway, when Norway > Sweden > Denmark in terms of alcohol tax. The Bergen-Oslo train journey is very beautiful and highly recommended, and Bergen as a town is full of holiday, seaside atmosphere with its old wooden buildings. An interesting round trip could be here -> Bergen -> Oslo -> Copenhagen (maybe with a stop in Gothenburg along the way) -> Hamburg - > Amsterdam -> here. If you have any more specific questions about these places, feel free to ask
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I try to take one day at a time, but sometimes several days attack me at once.