If you want to trigger the YN-565EX II from the YN-560-TX, you need to add a compatible receiver to the foot of the YN-565EX II. You can use (from oldest to newest):

• RF-602RX (the receiver in an RF-602 Tx/Rx set)
• RF-603
• RF-603 II
• RF-605

The RF-603 II or RF-605 units are the better choices today. The 602 and 603 have very awkward placement of the on/off button so that you can't reach it when a flash is mounted on one as a receiver. The 603 II/605 have the power button as a slide on the side. They also have higher sync voltages, can be used with mirrorless cameras, and have an overall better feature set for not much more money. The 605 adds group control.

However, be aware that this triggering will be manual-only, just as with the YN-560III. You will not have TTL or HSS with the YN-565EX II off-camera.

If you want HSS/TTL on both off-camera flashes, you will have to switch the YN-560III for an HSS/TTL capable flash and use a TTL-capable triggering system.

Today, I would recommend starting out with Godox instead of Yongnuo, because for roughly the same price, you get a lot more function and better expansion options, and particularly because their TTL and manual-only gear uses the same radio triggering system and work together.

See also: What are the Yongnuo flash naming conventions?