How to pick a tubed integrated without hearing one

Try a late model Jolida 102B. You'll be amazed by what 20 EL84 tube watts can do. Then you have the fun of tube and cable rolling on the cheap. Your room seems small and asymmetrical: it should work well.

I second the Cayin, great sound, great value.

If your speakers are reasonably efficient and you're not trying to fill a large space the Jolida FX10 EL84 based integrated is great sounding, comes with a remote and is auto-biasing.

Sophia Baby Electric may well be the greatest buy in all of Audio,past or present.

As other comments suggest, the efficiency of your speaker will play a big role here. I'd choose a speaker first (keeping in mind that you want ones that pair well with tubes). For speakers, you will want to think about whether you want very high efficiency ones that play well with SETs and what are known as fleawatt amps (a great budget example is the Miniwatt N3 or the S1), or a speaker that needs the 10-50 or so watts that a push-pull design can give. Most bookshelf and monitor speakers are of the latter type. Many in this design have quite low efficiency.