Scroll down to the bottom of the MKL02 page, until you find the "Troubleshooting" section:
https://www.pjrc.com/store/ic_mkl02.html
The most common problems and solutions are explained in that section.
The very first troubleshoot step is the measure the voltage on the Program and Reset pins. If the MKL02 chip is running, it will turn on the internal pullup resistor on Program, so you should see 3.3V on that pin if the button is not pressed.
A completely blank MK20 chip will repetitively watchdog reboot. With a voltmeter, you will see a voltage like 0.1 to 0.3V. With an oscilloscope, you'll see it's actually pulsing. If the MK20 has a normal program, you should see 3.3V on Reset.
If the MKL02 is running properly, it should detect when Program is low and pull Reset low. If the MK20 is blank, you should still be able to observe 0.1V become 0 volts. This simple test can at least confirm the MKL02 is alive and running.
Many things can go wrong. If you are completely stuck, please read that Troubleshooting section and click the links to the forum conversations. On this forum, you can see how other people perceived the problem before it was diagnosed and understoood. Maybe some of those experiences of other people can help you find a path to fixing your design?