Using heat-conducting permanent glue is the easiest solution for mounting the LED strips on the frame, just keep in mind that it will be difficult to remove the strips afterward without damaging them.
Before applying the thermal glue make sure both the aluminum and the LED strip surfaces are clean, free of any dirt or oil by wiping them down with isopropyl alcohol. Next, apply a generous and even layer of thermal paste on the back of the LED strip. Do not use too much paste as that would flow out on the sides. Now position the LED strip at the center of a cross bar and place it on the aluminum, use small back and forth movements to push out any air bubbles and form a tight contact between the bar and the strip. Avoid touching the LEDs or getting thermal glue on them, if you do get some glue on the LEDs remove it with isopropyl alcohol and a soft toothbrush. You have about 5 minutes before the glue starts to get thicker, it takes a total of 2 hours for the bond to fully form. If you ever need to remove a strip, try using dental floss.
Note: In some photos, you will see the strips mounted with screws. This was the old way I mounted my strips and it was a horrible job to drill 50+ holes. Since I moved to thermal glue I never looked back.