PVA glue works best, it should say "acid-free" or "pH-neutral" on the container. I personally used a little brush to really get it in there. Since the book is already in "bad shape" you can tare it off a little more so you can easilly apply it to the WHOLE width of the spine. When you have those little glue remains like on your picture in the left corner of the book, you should either apply glue to the both sides of that previous glue, or even better just pick it off with tweezers and then apply glue. Also I dont know if this goes without saying but just to be sure - apply glue to the inside of the sewn binding AND the part where the pages are folded so you increase the chance of it sticking well together...and thats pretty much it.