Is there any way I can get them without having to reinstall the whole thing?
If you dont have any backups for your database then the answer is - NO.
Even if your image files are intact, the information about them is gone (based on your original post), so it means you will need basicaly start over.
I know it will be very paintfull but I'd suggest you follow these steps:
1) since it seems you already created the database and all needed tables, you dont need instal.php anymore, now go to ACP (Admin Control Panel) -> Settings and check that everything is the way you want to be.
2) Since information about your categories is gone, you will need recreate all categories manualy. But, there is a little trick - you have all your images already placed in data/media/XX/ folders (where XX is a number - category id) So, even though you recreate all categories from ACP, it does not mean that the images will be placed in the correct categories, because category IDs might not be the same as they were on your "old" site...so, you will need move the images into proper XX folders...well, ofcourse if its easier you could simply rename the folders the images in to the correct name as the new category id...like do it one category at a time: create category, find out it's id, find the files that are belong to that category and rename the folder they are in...
3) once you recreated all categories and move all files into correct folders, you will need run "check new images" in ACP and add all images into the database...
Paintfull, but perhaps this will help you to do database backups periodicaly
Good luck.