Could I use the new LED lights in my current kitchen ceiling fixture

It looks like you need to upgrade. I hope this film helps you.

https://www.youtube.com/watch?v=Ccf5n92NhSk

Yes you can. They have LED bulbs that look like incandescent bulbs. They screw in just like old style bulbs. One thing to pay attention is LED bulb brightness is based on Lumens and not wattage. Packaging will state the equal wattage comparison to incandescent bulbs. Rule of thumb is lower Lumens will give a softer warmer light. Higher Lumens will give a brighter almost daylight light no matter what the equal wattage is. Basically LEDs up to 950 Lumens will be a warmer light. 1000+ Lumens will be a brighter (almost daylight) light.

William, you are mostly correct. A second number to look for on the package besides lumens is color temperature. A color temperature around 3000 degrees Kelvin will yield a warm white regardless of brightness. The higher the number the bluer the light and around 5500 degrees is daylight, too harsh inside for most people's taste.

I had all CFLs in my house but have been upgrading to LED whenever one quits working. LEDs use slightly less power, not enough to notice, but they are instant-on unlike the reflector CFLs in my shop that take nearly 5 minutes to come to full brightness.

Go for it Rose. It is a great idea to change to LEDs. Much less power so you won't feel guilty if you forget and leave the lights on as well as probably never having to change the bulb again.