Its a slippery slope upping budgets but if you creep nearer 200 then you could buy the 1st gen 2nd hand RX100 by Sony. A spectacular camera irrespective of format!
Otherwise a slightly different recommendation which was panned by many when it came out, but since a firmware update was pretty good- The Canon EOS-M. It comes with an APS-C Sensor and is compact with the 22mm pancake, the bodies go for around 130, or with a lens just north of 150 if you search on Ebay. I have one and gave it to my sister who loves it. Touchscreen AF etc. Another league compared to the compacts, if the size isn't too big for your wife.