A couple things can cause this:
- If you are using a mix of 5.25" and 3.5" external drives, make sure the 3.5" drives are connected FIRST (i.e. closer to the computer) on the Y-cable.
- The only 5.25" drive that's compatible is the "Apple 5.25 Drive" (model #A9M0107). You cannot use the "UniDisk 5.25" (model #A9M0104), even though it looks pretty much identical on the outside. Both drives use the same 19-pin connector, but the UniDisk 5.25 requires -12 volts, which the Macintosh LC doesn't support. You can look on the bottom of the drives to find their model numbers.
- The metal frame around the 19-pin connector on the drive's cable can sometimes get bent at the corners. That can keep the plug from connecting firmly, which can cause the IIe card to not see the drive. In any case, I'd suggest unplugging the cable, checking that all the pins are straight, and ensuring you carefully re-plug it in and re-tighten both screws.
I had exactly the same symptoms as you when I first set up my own IIe card in an LC475. In that case, everything looked fine and I knew I was using the correct model of drives (I had two Apple 5.25 Drives daisy changed). But the computer would simply ignore the drives completely. Turned out I wasn't as observant as I thought, and the second drive on the chain was actually a UniDisk. Darn those identical cases.
If none of the above ideas fix the problem, I'd suggest swapping in a different Apple 5.25 Drive and different drive cable to test.
Good luck. The IIe card (when you get it working) is a real treat, and since the Mac LC can use a cheap VGA adapter, it's one of the easiest ways to get a "real" Apple II experience on a modern monitor or big TV.