Launching the latest backup does not overwrite the original catalog. Use File->Open Recent. (Or, you have to find it on your hard disk—it's name ends with .LRCAT and open it)
You need to reconnect the catalog to the photos on the MacBook Air HD, hopefully the folder hierarchy is the same, in which case reconnecting is easy. Please follow these instructions Adobe Lightroom - Find moved or missing files and folders (see Figure 4 and the text following)
Take the Physical hard drive out of the external enclosure and connect it to your system with a SATA to USB adapter of some type, one that has its own AC power supply, and see if you can read the drive.
99% of the time it is not the physical, actual, HDD that fails. It is the enclosures electronics that fail. Removing the drive from the enclosure and connecting it to a system usually allows you to get the data off it.
I will try to reconnect the catalogue to the photos in the hard drive of my Macbook Air, the possible problem is that I have changed some names of the folders in the ext HDD; the folder hierarachy may be a problem as well because I have taken more photos after I broke the ext HDD, and added to the hard drive of my Macbook Air.oot
And Shootisbond, if the enclouseR of the ext HDD is opened, the warranty will be forfeited; it isn't extremely large amount of photos as I just used it since March, hopefully I can reconnect the catalogue to new location of photos, then I still have warrnty of the ext HDD, it could be fixed or replaced.
Thank you for your suggestions.
Might I suggest that in the future, that all copies and backups of photos maintain the same file name and folder name. As soon as you start changing your folder names on one copy/backup but not the other copy/backup, you are causing yourself problems.
Better yet, in my opinion, don't use folder names to organize at all. Use keywords and other metadata, keep the folder hierarchy as simple as possible (I use a Lightroom default which makes folder names to be the capture date, and this takes zero effort on the part of the user), and I never change folder names or file names.