How come he didn't use the most obvious solution to "grasp" time - dates?

I don't think Lenny really cares how much time has passed. He's on a quest for revenge, whether that takes a month, a year or a decade. He intends to keep going until he succeeds, so while the dates may be useful in establishing a more concrete timeline, they aren't going to make a difference to the one question he really cares about - "is it done?"

Of course, given Lenny's condition, dates are not really any more reliable than the other records he keeps. If his notes, polaroids and tattoos are all theoretically open to sabotage (especially by himself), then why wouldn't the dates also be open to similar sabotage? Handwritten dates on paper or polaroids could be erased, or recorded incorrectly. Perhaps the method he uses to establish the date could be unreliable (ie: someone, possibly himself, edits the police report to change the dates.)

Dates in tattoos might make him realize he's been on the hunt longer than he would otherwise be aware, but again, he doesn't care how long it takes, only that he finds his revenge. Knowing a particular clue is X years old would only make him more desperate and motivated to find SOMEONE he can kill and call it "justice." And, of course, if he is sabotaging his own progress in order to continue killing, this desperation would get stronger over successive loops, but it wouldn't likely stop him from trying.

Speaking more generally about the movie as a whole, I think a lot of this is intended as commentary on human motivation and the dichotomy between belief and fact. People tend to believe what they want to believe, even in the face of factual data that contradicts that belief. They can sometimes come up with some truly mind-boggling excuses for why the offending facts should be disregarded. This is especially true when a person's beliefs are strongly tied to a goal or driving motivation that they consider to be a sort of "life's work." Admitting the flawsin their belief then becomes equivalent to admitting their purpose in life is somehow incorrect, which is something very few people are actually willing to do.

Lenny's quest for revenge is an extreme example, of course. Movies use extreme examples of things because they tend to make for better drama. Lenny's quest has become his life's purpose - to find and kill the man who murdered his wife. But once that job is done, he no longer has a purpose. This, coupled with his memory condition, is what leads him to continue the cycle. So that he still has a reason to keep living, even with his crippling mental condition. Given that, I don't think the use of dates (or any other 'solution') would be sufficient to stop him. He'd either find an excuse to doubt the dates and thus stop trusting them, or he'd come up with progressively more insane theories to explain what "must have happened" using the dates he has, given whatever the current date is.

To add to Steve-O's excellent answer;

Keeping track of time would be the worst thing Lenny could do. His entire world is anchored in his wife's death, he forgets everything after it, and rarely brings anything up that happened before it. His entire motivation is revenge for his wife's death, and the only way he sustains himself is by keeping that anger fresh - it fuels every interaction he has.

If he ever found out he'd been at this for years without a single success, it's all too possible to become demoralized. He'd question his purpose, his existence. If he hasn't found the killer after all these years, why would he find him now? Is he on the right path? He might even go in circles, trying to find out why he hasn't found the killer - all of these things are completely unproductive. Lenny doesn't want questions, he wants answers.

So the question isn't why doesn't he keep track of time, but rather why would he?