Dioder strip was not designed to fit in Detolf, so you are already trying to do something starting with the wrong parts.
Let me put it clearly:
You have bought absolutely the wrong lights for the Detolf cabinet and your absolutely best option is to return them for a refund and use something else.
Now you have 3 options:
Buy the correct part from Ikea: If you want the only plug-and-play LED for the Detolf from Ikea it's a round single module that you can insert into the hole on the top of the unit. Illumination at the bottom shelf will be poor because there's only a single light source at the very top.
Buy something else: If you want to buy something else, you should look at LED ribbon lights from Amazon or eBay with a suitable power transformer (my recommendation). Very versatile because you can use as little as 3 LEDs to make your own small strips. This lets you create dramatic lighting instead of blasting each shelf which makes your display look like a sandwich refrigerator in a cafeteria.
Use the wrong parts: If you want to use the wrong parts you already have, you must modify them. I looked at the instructions and photos online and it's a very simple task to check to see if the MALE connectors at the ends of the wire leads will fit into the FEMALE receptacle on the LED strips. If not, it's also very simple to use the correct male plug end from the smaller adapters. Cut them off, cut off the plug from the longer wire leads, and then attach this new male plug to the longer wires. Wrap with white electrical tape or white heat shrink tubing.
That's it. Those are your only options, aside of course from having the ugly mess shown in that other image, which as I understand it, is something you're trying to avoid.
What I can't understand however is someone asking for help, getting a ton of it, and at each step along the way putting up roadblocks to shut down that advice instead of continuing the conversation to get the job done.