Hi Olivier,
Sorry for my later reply because of China's National Day and Mid-Autumn Festival.
It seem your hardware have problem.
1. Check the SWD interface, whether it is correct with the LPC LINK2,
This picture is the LPC LINK2 SWD interface:

Check the connection with your board.
Normally, the SWD interface need this pins:
pin1:3.3V.
pin2:SWD_DIO, this pin you need to add external 10K pullup resistor in your board.
pin4. SWD_CLK, take care, you don't need to add external pull up, even you add, you just need to add the pull down.
pin10: reset pin
pin3,5,9: GND, connect to your board GND.
2. Use the JLINK firmware to check the core
If your point 1 is really correct. Then check this part.
1) download and install the JLINK driver
SEGGER - The Embedded Experts - Downloads
2) Change your LPC-LINK2 to JLINK firmware
Download and install the LPCScrypt
LPCScrypt v1.8.2|NXP
power off the LPC LINK2, disconnect JP1, power on the board again(J5 USB port), you will find you board is the LPC device, it is the DFU mode.

Open the jlink in LPCScrypt:

Press any key, wait the jlink firmware download finished:

Then power off the board, short jp1, power on the board again, in control panel, you will find you board is JLINK.

Open JLINK command in the JLINK driver install path:

Can you find Cortex M3 core?
Please check this at first, if you can't find it, you need to check your LPC board, your hardware have problem.
Have a great day,
Kerry
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