Attacking with an active Ring of the Darkhidden does not break the limited invisibility it grants.
Nothing in the ring's description says that attacking breaks the limited invisibility it grants. Importantly, it does not include the phrase "as the spell" (thanks @fectin for mentioning that) common to a lot of "spell in a can" magic items - notably including the Ring of Invisibility, which the wearer can activate to "benefit from invisibility, as the spell".
The "invisible" status assumes that attacking does not break invisibility:
If an invisible creature strikes a character, the character struck still knows the location of the creature that struck him (until, of course, the invisible creature moves). The only exception is if the invisible creature has a reach greater than 5 feet. In this case, the struck character knows the general location of the creature but has not pinpointed the exact location.
The spell invisibility explicitly ends if the creature it renders invisible attacks, further cementing that "attacks while invisible end invisibility" is a specific "feature" of that spell, rather than of invisibility generally.
Since "attacking breaks invisibility" is a feature of the spell invisibility (and a few others that are "invisibility, but..."), there's no particular reason to believe that attacking while wearing the Ring would render the wearer visible to darkvision.
"But", you say, "the item is based on the spell invisibility! Surely that means that attacking ends invisibility." The requirements on magic items serve a couple of purposes (eg., gating who can make the item - a Ring of the Darkhidden requires invisibility, which is an arcane spell; clerics would have a hard time making such a ring). However, the effects of the spell and the effects of the item aren't firmly linked - see the Ring of Communication (also on p122 of the Magic Item Compendium), which requires detect thoughts, but grants that:
When you activate a ring of communication, for the next minute you can clearly hear any words spoken by anyone within 1 mile who is wearing another ring of communication to which you are attuned.
The Ring of Communication does not grant telepathy of any sort, instead relying on spoken words, even though the base spell is detect thoughts - one of the earlier telepathy spells.