Butting a skirting board to another at 90 degrees

I have boxed in some pipes in my bathroom by making a false wall a few inches forward from the existing wall. All the existing walls have skirting boards.

I now need to butt a new piece of skirting board at an angle of 90 degres against the existing skirting board which continues into the hidden boxed in area.

I can't figure out if it is possible to make this cut with a mitre box. It seems like the lower part of the skirting needs to be cut straight and the upper part with the curve needs to be cut at 45 degrees. I do have a similar piece of skirting board which has been cut by tne previous owner, but if looks like it may have just been cut by hand and eye.

It's tricky to explain but hopefully this makes some sense.

Appreciate any suggestions.

Thanks

Sounds like it needs scribing in like doing a corner on skirting board but can you post a picture?
Edit:
http://wiki.diyfaq.org.uk/index.php/Scribed_Joints

Thanks lostinthelight, I think your link describes exactly what I need to do.

I can't quite get my head around Step 2, and cutting the shaped part with the coping saw but I have some spare skirting so will practice on it tomorrow and if necessary read up a bit more on scribed joints.

I'll also post some pictures in the morning.

Since you have existing skirting in place the new section will require the female cut.