How about saying what you do have available.

Are the files in the profile inaccessible as well, because if they are not accessible you will have to rely on your own backups.

Profile Files

• Profile folder - MozillaZine Knowledge Base
  http://kb.mozillazine.org/Profile
• Creating a new Firefox profile on Windows - MozillaZine Knowledge Base
  http://kb.mozillazine.org/Creating_a_new_Firefox_profile_on_Windows
• Transferring data to a new profile - Firefox - MozillaZine Knowledge Base
  http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox
• Recovering important data from an old profile | How to | Firefox Help
  https://support.mozilla.com/en-US/kb/Recovering%20important%20data%20from%20an%20old%20profile
• Profile folder - Firefox - MozillaZine Knowledge Base
  http://kb.mozillazine.org/Profile_folder_-_Firefox

Also in your profile besides places.sqlite which contains both bookmarks and history is a folder of .json bookmark backups that can be used to restore (not import) bookmarks.

I have access to all the files in the profile. I simply can't boot into the OS which the old profile is installed on. I just don't know how to import the files being that I'm not able to do an export (or at least I don't know how without being able to run the old installation).

I still need help. So I get the places.sqlite from the old drive, I still can't import.

Import/export only applies to bookmark html files. You can't export using the Firefox on the broken laptop. You copy copy a JSON file from the bookmarkbacks folder though but then all you would be getting are bookmarks, not history.

You have access to the drive, use XCOPY in a command window, or copy folder files from the external directory to your own directory or to the directory on an external backup drive by dragging and should say Copy not Move when you do so by holding the Ctrl key when you drag from one Windows explorer window to another Windows explorer window.

Windows 7 Home Premium -- File Transfers & Backups (#ftp)

http://dmcritchie.mvps.org/windows-7/win7.htm#ftp

Firefox Portable (notes)

http://dmcritchie.mvps.org/firefox/firefoxportable.htm#batfile

Backup your files, always take backups

http://dmcritchie.mvps.org/excel/backup.htm

You can copy the file places.sqlite from the profile folder with the not working OS to your current Firefox Profile Folder

• http://kb.mozillazine.org/Profile_folder_-_Firefox
• https://support.mozilla.com/kb/Recovering+important+data+from+an+old+profile
• http://kb.mozillazine.org/Backing_up_and_restoring_bookmarks_-_Firefox

• XP: C:\Documents and Settings\<user>\Application Data\Mozilla\Firefox\Profiles\<profile>\
• Vista/Windows 7: C:\Users\<user>\AppData\Roaming\Mozilla\Firefox\Profiles\<profile>\

The "Application Data" folder in XP/Win2K and the "AppData" folder in Vista/Windows 7 are hidden folders.

• http://kb.mozillazine.org/Show_hidden_files_and_folders

You can also access your current profile folder via:

• Help > Troubleshooting Information > Profile Directory: Open Containing Folder