Do you mean connect 2 separate runs to 2 outlet? I put far more research into this for you than I reasonably should have at work haha.
12v LED strips struggle after running over 16ft, so depending on the size of your kitchen you will want to either get a 24v system with a split, or 2 basic kits and a plug adapter. Make sure you get water proof strips that will resist the heat and splashes in the kitchen.
Connections This kit has all the connectors and fasteners you need. The primary being the DC splitter. You also definitely need the screw brackets. No matter how good the adhesive is, it won't hold long. A bonus is the extra length connectors allowing you to get around corners nicely. The white end connectors are super easy to use; just cut along the line on the strip and clip on the connector. this way you can turn 1 roll into many lengths. (remember, power draw is based on total amount of LEDs. 1 length of 16ft is equal to 4 lengths of 4 ft)
DC power extensions This will get your lights out of the cabinet easily without wasting lights and effective length.
Smart plug No matter which route you choose to take, I recommend using a smart plug adapter. You have many options to activate them, such as you alexa/google/ect, phone, light switch motion detector (my favorite), or a bunch of other options.
I can't really tell what you've got going on there. The ending looks like it MIGHT be a push-in ribbon connector. They're usually 2 parts -- you open it, push the two together, then close it. That would be the blue circled thing I'm guessing. That gray bit may be able to be pushed away from the strip, but it's hard to tell from image.
However, the long & short of it is that you've got basically 2 wires there in the LED strip -- a positive and negative. You can see the labels on the strip (highlighted in my image above). You can also see the copper connectors and the line where you can trim the strip to length. As long as you get the power to the similar connectors under that circled connector (or to the next set of connectors that I've highlighted) you'll get light. (If you skip to the next set of connectors, the skipped LEDs will not light but the rest will.) You could just solder some speaker wire from the green connector over to the corresponding spots on the LED strip, or buy another LED connector to do the same thing. Something like these (but check the size) https://smile.amazon.com/HitLights-Connector-Include-Solderless-Splitter/dp/B07XDVNDZQ
Hope that makes sense.
So where do I get the male power connector for those extra strips that come with it? Also kind of confused how these strips work with little white connector pin things on the end. I guess it's meant to string together.
I had ordered this connector kit for 3528 or 5050 type lights. I'd prefer a cool color:
https://www.amazon.com/HitLights-Connector-Include-Solderless-Splitter/dp/B07XDVNDZQ
And was going to use lights like this:
https://www.amazon.com/Flexible-6000-6500K-Waterproof-Lights-Bright/dp/B00X5MEHGS
As far as I can tell, you cut the strips where you want and use the attachments from the connector kit. Would that work?