No, it only effects the narration. On a failed saving throw she still tells the truth. Because the 'truth', mechanically, is that she doesn't have the information. THAT was a narrative not a mechanical change as well.
As long as you acknowledge that the narration in both cases is just fluff, and doesn't actually change the underlying facts of the game, everything stays consistent with RAW.
This requires a house rule on reality and the definition of truth, though. The play is that the character DOES know the real truth, but is not revealing/acting on it because of a narrative choice in dealing with a failed save. The character believes they know the truth, and so an answer of 'I don't know' is a lie from the point of view of the character.
Zone of Truth doesn't check to see if statement made are consistent with previous mechanics rolls, it checks to see if the truth, as understood by the characters, is told. If LOL believes she knows the answer, then any answer of 'I don't know' rings as false. To patch this, you grant the ability to resist the spell after a failed save and lie because the patron extends magical protection to protect this information. In other words, you institute a house rule that allows a player to ignore the mechanics of zone of truth to protect their narration. And that's fine, but you're bending and breaking the rules as written, and not just soft ones like how INT is defined* but hard ones like how ZoT functions.
* On which I agree with Max: it is a rule because it's a definition of a class of thing in the game, even if it doesn't have a number attached to the definition. You're welcome to change that definition, but, in doing so, you're instituting a house rule. This is easily checked by considering a new player reading the rules -- they will normally think that INT being defined as it is means that's the rule for what INT is. It's only us experienced guys that have a few systems under our belt and an idea of how we like to play that would ever look at that and think 'oh, that's just a suggestion, I can change that and it's not a houserule.' But, it is still a houserule, even if it's one that you don't think is a very big one.