Unfortunately, since your old Firefox data is stored on your now deceased hard drive, you won't be able to restore your old data.

If you were using Firefox Sync on your old laptop, you may be able to sign into Sync on your new laptop and recover some data that way. However, since Firefox Sync is not a data backup service, it's not likely that you will be able to recover all (or any) of your data that way.

However, there may be some hope. Over my many years working with computers, I've been punished many times by a "dead" hard drive. However, in many cases it turned out that the hard drive wasn't actually dead and could still be recovered. A lot of the times it's the formatting of the drive (and issue that happens a lot with Windows) that makes it so that Windows cannot boot and Windows cannot access the information in it. Sometimes something as simple as just plugging the drive into a Linux computer will work.

If you are able to bring it to a computer store (if you are not done so already), I'd recommend doing so. Some are willing to tell you whether or not they will be able to recover any data as a free consultation before billing you for the recovery. Obviously, if it's something quite easy, you likely won't end up paying a lot, but data recovery can get quite pricey. So, if it's really expensive, it may be more economical to start from square one again.

Hope this helps.

That "dead" hard drive may not be completely inoperative. Many times a hard drive that won't boot the operating system is usable as an external device when connected via a USB port; at least long enough to retrieve user data.

If you remove the old hard drive and connect it to the new laptop thru a USB adapter cable such as this you may be able to recover your Firefox bookmarks.