There isn't really any U.S. equivalent to the Home Secretary, the responsibilities are divided over several offices reporting to the president.
Additionally, the Secretary of State is foreign affairs and not the equivalent of the Home Secretary.
https://en.wikipedia.org/wiki/Katherine_Archuleta
That is the person who is responsible for the OPM.
It operates on a fee-for-service model for 90% of its budget and isn't properly funding IT and IT security. Given she has only been there 18 months, I'm not sure how you can really blame her beyond the fact she didn't follow the "shutdown and repair" recommendation...except for the fact if she did that, she'd have to burn through the OPM's cash reserves as the only way to fund it.
Its honestly more of a flaw in the way Congress built the OPM incentive structure and the fact the director is a revolving door position that changes every 2-4 years.
The government simply just doesn't fund/prioritize IT & IT security properly outside of the intelligence services. They also build these psuedo-private-sector funding models that create the incentive to "ignore" cost centers like Security into critical infrastructure like the OPM. Its really horrific.
It's a long time since any UK politician resigned on principle or on honor. I think the late Robin Cook was the last (over the Iraq invasion which he believed was illegal).
Private Eye has an endless stream of 'highly suspect' (since any suggestion of 'corrupt' may get HN into trouble, just to be clear, I'm not saying 'corrupt'.) activities of various government ministers, none of whom resigned or even came close - despite their misdeeds being exposed. Look at how many were caught in the expenses scandal - some UK politicians even tried to hold on while being convicted of fraud and other criminal convictions (with some initial success too, https://en.wikipedia.org/wiki/Nazir_Ahmed,_Baron_Ahmed). That's never going to fly in the US.
You really need to go back to Margaret Thatcher's cabinet before you can really find many examples of resignation on the back of government screwups.
I think a resignation is more likely in the US than the UK. And it's not at all likely in the US.
The end result of honor resignation is that you have politicians who worry more about appearances than actually getting anything done. Does that sound at familiar?