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Good job in using the native port to program it. I absolutely despise the programming port and whoever decided to put it on the Due should be slapped silly.
The JTAG port can be used to program and debug the SAM3X chip. The pinout for JTAG on the Due is standard. A SAM-ICE has a large header but there is an adapter to go from that to the smaller plug on the due. This is a standard thing and works fine. Hopefully someone can link you to the adapter, I forget what it is called.
A SAM-ICE works fine with the Due for both programming and debugging. I've used it for both uses. It all integrates into Atmel Studio.
The complete JTAG-ICE has the correct cable included, but it is quite a tight fit on the Due, something else is interfering with the edge of the connector, but it goes in, just-just.
Atmel Studio 7 directly supports the ICE for both programming and debugging.
Atmel-ICE. It supports both ARM and AVR.