Hello Usman, I am by no means an expert here but I will offer my thoughts for you. I understand your desires on keeping it in the full frame line to use it as a backup. I also agree it will be difficult to fit within your budget a full frame and lens. Have you thought about a used body? I have purchased a used lens and I am quite happy with it.
That said, I have a D5100 and it is a remarkable camera. Again I am a beginner with only 1 year experience with the D5100 (and DSLRs for that matter) but this camera has quite a bit to it. With the exception of not meeting your full frame requirements I think your wife would like the camera. Also, I believe you can use your full frame lens with the DX body.