Adaptor for USB Blaster II to USB Blaster header

Well, I have measured all the 2x5 100mil headers I have on all my different FPGA programming boards and the inside dimension of the boxed header comes out to be 700mil X 250mil. So the 'standard' 2x5 female header fits perfectly. I have third party USB Blaster clones, a TerASIC USB Blaster, and an original ByteBlasterII (ie, LPT port to 10pin header). All are exactly the same female 2x5 connector.

Both the 2018 and 2022 versions of the Altera/Intel USB BlasterII user document call out the connector as 240mil x 700mil:

So I would postulate that your board has 2x5 pin male shrouded header that is 500mil wide, expecting to be used with
a 500mil wide female header, as you might see from breakaway header strips.

You will either need to modify the board and replace the non-standard connector, or if that is not possible due to mechanical
fit issues, build an adapter cable. Source a female 2x5 flat cable connector that fits in your socket (if you can ... IDK) and then
attach it to a short 10wire IDE cable with a 10pin box header male connector that is 'standard size' of 250x700mil.

Or make a little PCB with the two 2x5 100mil footprints and source a female board mount connector that matches your
board mount male connector and use a standard 2x5 100mil male boxed header as the other connector.

Or if you can't find female connectors that mate with your male boxed header, get out a dremel tool and make some from standard
female connectors by cutting/grinding them down to size.

And whoever designed your original board and used a non standard header owes you a real nice dinner...