I'm not sure if that Ikea unit would hold up. Given the dimensions of each cube, I would guess that the unit was designed with LP storage as one of the possible uses. But it may not have been intended to be completely filled. Hopefully, when they say that the capacity is 29# per shelf they mean per cube.
An average LP weighs about half a pound. So you would reach capacity at 58 records, which is about 10 1/4". Those cubes are probably 13 1/2" across. Ikea's a pretty sharp outfit, and can probably tell you how roughly many albums you can get in the rack.
That said, that unit is the best option I've seen in a long time. LP racks are few and far between. Have you considered checking the phone book for cabinet makers? Carpenters usually don't tackle this kind of stuff. The best option I've seen for music storage for the obsessive collector is
http://www.can-am.ca/. Other companies solutions become unwieldy beyond 200 CDs or so. Can-Am is expensive and even more so for LP storage. But they've done the weight calcs and have a lot of experience.