Things aren't progressing well with my dead hard drive, so I wondered if - worst case - anyone has data recovery places that they'd recommend, preferably in New York City.

The average cost really begins at $450 despite their "free estimates" or $279 flat fee claims. Is your damage at that value? If the plates are cracked or the magnetic material peeled off, or the bearings welded together, it still costs you $450 for no data rescued.
We recommend a place in Ontario, Canada as they best you can find.
Have you tried installing the drive in a USB enclosure that has its own power supply to see what you can see, or if the drive turns?

Some of these may not still be in business. Some data here is from my two-year old book, but these were good when last used.

Here are some we have used with good results when our clients needed a rush job for payroll or taxes or whatever.
Hard Drive Recovery; Free Evaluation, Fees start at $279, 877-840-3282
www.acsdatarecovery.com

Data Recovery/Repair, NYC; Fast www.AmnetDataSolutions.com
New York, NY

Hard Drive Recovery Experts; RAID, NAS, SAN server recovery. 888-567-8328
www.OntrackDataRecovery.com. Been in the business for at least 24 years; I think they are tied into the Ontario, Canada firm

Data Recovery New York; (212) 661-1145. Free consultation
www.ComputerAdvisorsNY.com

Hard Disk Data Recovery; Experienced, Reliable; www.MilesTechnologies.com/Recovery
New York, NY

Recover Data Quickly
www.DriveCrash.com $279 Data Recovery Flat Rate.
Has "Class 100" clean-room; corporate work as well.
www.aerodr.com

New York computer repairs; data recovery consultat free (212) 361-9981
www.BnyTech.com

Drive Recovery of New York; www.WeRecoverData.com/New-York/NY Hard Drives, RAID, Disk, Server; Free Est 212-594-5946

Data Recovery Service
www.DataRecoveryLabs.com No Recovery No Fee - All work done on site - 866-340-0111