Cross-stitch conundrum

Cross-stitch is one of my secret weapons for staying sane. It is like a litmus test - if I haven't had time to sit and cross-stitch in recent weeks I know that I've been way too busy. I know that this HA hasn't been all too bad as I've completed one cross-stitch (the first one pictured) and almost finished another large one (not pictured).

However, I do have a problem. I have a largish cross-stitch that I completed in 2006 (I only know this because I stitched the date on it at the time). It was a kit given to me by my parents to do for my sister. Complicated!

More complicated than that is that after I finished stitching it I posted it to my sister (we were in Japan that year and she lives in Australia). I organised for a friend to frame it for her. However that friend (who reads this blog - don't feel guilty friend!!!) found life got way too busy on her, and wasn't able to frame it. She returned it to me last year and I haven't managed to get it framed either. Actually, I took fright after I had two birth samplers framed last year by a professional (see the photos). They look fabulous, but were frightfully expensive to have framed. So, I thought - I'll do it myself.

Here is my problem. I've bought a suitable frame with a matt the perfect size (see third photo). However the Aida cloth isn't large enough to overlap the edges of the backing. What to do? I know that to do it properly I shouldn't use the cardboard backboard provided with the frame, I should use acid-free board. But again, I'm trying to do this on a budget...my only thought is to sew some cloth onto the sides of the Aida to make it suitably wider and longer and then lace the back. I'm missing my sewing machine more than ever (it's in Japan).

Any cross-stitch framing experts with advice out there? Remember, I have less than 10 weeks...