I just visited USpace today and got to know Brad, the owner, a little better... he's really chill and cool, and you can definitely catch him on Wednesdays! The place has a good selection of mid-range to high-priced ukes - I remember at least one HF-3, one HF-2, a Pepe Romero Cigar Box tenor (handmade by Pepe Romero, at least $2k o.o), a KoAloha tenor, super soprano, concert, a D-VI, a Covered Bridge redwood/myrtle tenor, and some Kamoas and Kalas. He also has good coffee on hand as well! As for parking, just make sure you have cash on hand, because it is pretty tough to find a good parking space in the middle of LA!
If you're willing to branch out a bit farther than LA, Folk Music in Claremont (further east) has a pretty friendly atmosphere with a fair selection of ukes. They just had an ukulele band play not too long ago (the name escapes me, but they do a cover of MGMT's Kids) in their shop. McCabe's in Santa Monica (opposite direction) is also a good place to stop by, especially because they have concerts every once in a while (not just uke concerts, though). TBH, McCabe's has the best selection of high-end ukes (or maybe tied with USpace) - I saw an aNueNue harp tenor, two Blackbirds, a KoAloha super concert, some Collings, a HF-2, one of Pepe's 6-strings, and then some!
If you specifically wanted to try some Kanile'as, Kaye's Music Scene in Reseda has a lot of Kanile'as and Islanders, along with a Godin Multi-Uke... George, the owner, is very nice and the ukes there are not bad.
Maybe you should give the new ukulele you just got to your daughter and buy a new one for yourself in LA... just a thought... xD