Let’s start at the end first, the part where you say you are working from the previous designer’s Indesign file… Why are you using a 72ppi file from the client then? Is it something you are trying to place into the InD file?
As for the 72ppi, that’s it. You can’t “make” more pixels via interpolating upwards. They just aren’t there. What happens when you tel Photoshop to “make” it 300ppi is the program guesses based on neighboring pixels. That’s where the blur comes from. Once an image is saved to a lower resolution, that pixel information is gone forever. If you want this image to be 300ppi, it will be 2.76" x 4.08". If you wnt it to be 7x10.84, it will be 118ppi. Whether or not 118ppi will work for you depends on what quality level your client expects. When in doubt, print it out. If this is cover art though… I personally would expect better.
Here’s a basic presentation on image resizing:
DPI and resizing
It was written a while ago but is still relevant today.
This is very basic resolution stuff. It is one of the many things you should already have well under your belt before taking on clients.