Quis custodiet ipsos custodes?
There is no specific body or mechanism in place for the investigation of systemic equality and human rights violations in the EHRC.
It is general problem, and there is no simple solution. In this case the organisation could be investigated by the Government department that established it. There are civil servants who are able to carry out such an investigation if needed. Or a judge-led public inquiry could be held. And there is a minister who is able (with the support of Parliament) to make the decision to end funding.
The UN could investigate, but so could the Daily Mail. The UN has no oversight role in the UK government (or any government)
It is expected that such action is rare, which is why there isn't a standing procedure. It is actually quite rare that an organisation fails so far in upholding its own code that it needs this kind of action. Though your question could be asked about the HSE, and of course the police themselves.
Since it is rare, it is exceptional, and exceptional situations call for exceptional solutions. Compare this with the RUC. It was recognised that the RUC had systemic issues of trust, respect and authority with a significant part of the community which it was supposed to serve. This was systemic, and beyond the ability of an organisation like the IPCC to deal with. And so the RUC was, eventually disbanded by the government and replaced with the PSNI, following the Patten report.