Spent a good half hour looking for the manual online, only to remember Ikea manuals are all pictures. I found the manual and was reminded of this. About to give up, I took a risk, and thankfully bought the right sized key. Allen wrenches are made in both Imperial and Metric scales; Ikea, being Swedish, meant correctly picking metric.
I later looked through the manual, and on the 2nd to last page, it documented the correct key size.
https://en.wikipedia.org/wiki/VeloBind
>It is possible to take a soft covered Velo-bound book, remove the old binding and cover, and re-bind it with a hard cover, which may be pre-embossed for more a more impressive appearance. This rapid up-grade was the cause of the short-lived motto "Soft to hard in 30 seconds!" That was first done when the firm was located in Sunnyvale, California.