Puttering around on the Internet led me to this awesome discovery! You can make a paper battery holder that connects to LEDs with copper tape! This type of a battery holder is much THINNER!
I found the tutorial and a template, created by Jie Qi, at the following link: Paper Batter Holder Tutorial
Here is the project that I made using a piece of paper-cloth, two flowers printed from a 3D printer, two LEDs, and all of the copper tape that I had.
I just ordered more supplies, and I can hardly wait for them to arrive!