They look like the print! 
At least that's what Lab values are supposed to provide. What you ask for is the opposite of why most use color management. You've got a print and can see how it looks. In any illuminant. View it that way, that's what it looks like. You measured with a fixed process to gather Lab values and Lab is supposed to be based on your vision. At least the vision of a standard observer under a very fixed viewing condition. You are actually in a better position because you can see what that black looks like in all kinds of situations while the measurements report only one. The question I'd ask is what do you really want to gain with those numbers?
Now if you want to futz with driver settings before the profiling and try for the lowest Lstar values, which you could do visually as well (if you can't see it, does it matter?). Or you want to compare other driver settings with resulting Lab values, the numbers can be useful.
Hi Andrew - that's a question - DO THEY look like the print? What print? I interpreted Ben's question as presuming that the Black he was reading was SUPPOSED TO BE real true Black, say as read from a printer test target (Atkinson, Cooper, etc.) where Black is Black. If it were not supposed to be real, true black then you are right - the profile is probably providing the colour management it is supposed to provide, but we don't know that. He would be the best judge of it looking at the overall colour balance and luminance of his print under appropriate illumination compared with the softproof on his display.
Ben, I don't do i1Profiler (I have another device and application), so Andrew would be much better at addressing what to do about it than I would. That said, I presume you did calibrate the device before you measured the patches. Other than that, if the device is working properly and you are using a target known to be good for printer profiling and your profile-making settings are OK (illuminant etc.), I don't know what else to advise.
