Given everything you've already ruled out, the picture here points toward hardware failure rather than something fixable in software. Here's the reasoning and the one thing left worth trying.
Why Code 45 here means "not present," not "not installed"
Code 45 specifically means Windows still has a driver record for the device but the device isn't currently enumerating on the bus — it's different from Code 28 (no driver) or Code 43 (device reported a failure). You've already covered every software-side fix that could cause a false Code 45:
Driver reinstall/update (confirmed already current)
Full uninstall + rescan for hardware changes (didn't redetect)
Biometric service running
Full OS/firmware updates + Surface Diagnostic Toolkit
That combination — especially uninstalling the device and Windows failing to re-enumerate it at all on a hardware scan — rules out driver corruption, service issues, and most firmware/BIOS-table mismatches. If it were a software/enumeration glitch, the rescan would normally at least bring back a "problem device" entry.
The detail that matters most: no light
The fingerprint indicator LED and the sensor are on the same small board/cable assembly built into the power button. If the light doesn't illuminate at all anymore, that's not a Windows-level symptom — it means the sensor module itself isn't receiving power or has failed, or its connector/cable has come loose or failed internally. Software can't fix a component that isn't electrically responding.
One non-destructive thing left to try
Before concluding it's hardware, do a full power drain to reset the embedded controller (EC), since Surface fingerprint/power-button issues have occasionally been traced to an EC that's stuck rather than the sensor itself:
Shut down the laptop completely (not sleep).
Unplug the charger and disconnect any peripherals.
Hold the power button down for a full 60 seconds (this fully discharges the button's capacitors and resets the EC), then release.
Wait about a minute, then plug back in and power on normally.
Check Device Manager > Biometric devices again.
This is quick and has occasionally revived a "phantom" Code 45 on Surface devices where the sensor firmware initialization got stuck. It's a long shot here given the light is out, but it costs nothing to try.
If that doesn't change anything
At that point, yes — this reads as a hardware fault requiring service, not something resolvable through Windows. The power button/fingerprint assembly on Surface Laptops is a sealed, non-user-serviceable module (it's fused with the keyboard deck), so there's no user-level part swap or firmware "force redetect" that reaches down to a physically dead sensor. Your options would be:
In-warranty: contact Microsoft Support for a repair/replacement (this is a defective component, not misuse).
Out-of-warranty: Microsoft's paid Surface repair service, since third-party repair on this specific assembly is uncommon due to the sealed keyboard-deck design.
Windows Hello with a PIN, or a compatible external USB fingerprint reader, would be reasonable workarounds in the meantime if you want biometric sign-in restored sooner than a repair turnaround.