What happens if the electronic chip on your passport is broken at a port of entry to the US under the Visa Waiver Program?

My passport which is an EU one and has got the chip, but it has never worked yet. When landing in UK I still always choose the ePassport gate regardless. It takes about 2 minutes for the gate to realise that it can not read my passport, then person responsible for overseeing the gates will direct me to see the border control officer who will scan and examine the passport himself/herself. This is still faster than waiting in the line of people without the ePassport even though I always get rejected.

The regulation for the VWP is that you have a passport with a chip, not that it works. Chip could have broken on the air plane by cosmic radiation or other freak accident, yet people on the ground are still be able to verify who you are. Chip is just for your convenience and also employing less people at the airport and opting for gates instead, it is by far not your last verification resource.

Also I have travelled to US on VWP with that passport without any problems.

The USA only started to install software to read chip data in the APC kiosks at select airports a few months ago, and first equipped CBP officers' desks with similar software even fewer months ago at JFK (see this article/press release), so it might be a little early to have heard what happens when the passport chip doesn't work or the facial recognition finds a mismatch. I think it is a fairly safe bet that a failure of the process will likely just signal the human you are or will be talking with to look harder at you and your travel document than they might have otherwise.

If they were going to deny you entry for a non-working chip they would be telling the airlines that to give them a chance to deny you boarding for the same reason and avoid having to fly you home.

The Global Entry kiosks allow entry without talking to a human and they accept a variety of documents including chip and no-chip passports, green cards and sometimes NEXUS cards. Since they make no use of biometrics stored in the travel document (only the chip passport has any) but instead securely identify the person offering the document by comparing to centrally-stored fingerprints, they have even fewer reasons to be concerned about a not-working chip.

I in fact tried that experiment on my last entry using a GE kiosk by leaving the back cover of my passport, where I believe the chip is, inside the foil envelope I carry the passport in. The machine didn't notice.

If you want to test your chip yourself you might look for NFC tag reader software for your smart phone. I've used this app and tried others. If you can read the chip that way you can be sure it is working, though a failure to read it may be telling you as much about your phone's crappy NFC hardware as it is about the chip. Note the weak bit of NFC isn't the chip itself but the physically large loop of fine wire or foil it needs as an antenna to pick up enough energy to power the chip. While this seems pretty reliable when embedded in hard plastic, as in a credit or transit card, it may not work as well in a bendable cover of a passport.