From the wayback machine - found the link above...
https://web.archive.org/web/20141023230925/www.lpcware.com/content/forum/lpc1114-ssp-tx-fifo-clear
It appears that the SSPx in many LPC chips is an ARM Primecell PL022. There are test registers, which can be used to read the TX FIFO, thus could be used to flush it.
Since the link is no longer available... in brief, the solution described is to reset the SSPx module via LPC_SYSCON->PRESETCTRL:
LPC_SYSCON->PRESETCTRL &= ~(0x01 << 0); LPC_SYSCON->PRESETCTRL |= (0x01 << 0);
(Then reconfigure the SSPx after that.)