This is going to be LOOOONG, but informative!
My second of four daughters was like this... JUST like this. We were trying and TRYING to get her to go during the nighttime, but she was just a very deep sleeper. We had pull-ups on her @ night for 3 years (she was day-trained @ 2 years old)!!!! Her doctor(s) kept saying that it is normal and OK. Our research proved the same- that it is normal.
Well, about three months ago my husband and I REALLY tried HARD to get her to stop. It started with the pure fact that we were sick of buying pull-ups! While we started this process we used a protective plastic/vinyl cover under the sheets to protect the mattress.
First we told her that if she kept her bed dry for two weeks solid, she'd get whatever she wanted within a certain price (age appropriate, of course). That DID seem to motivate her, but we still noticed that most nights she was still wetting the bed. She seemed to be more interested in NOT doing it, though.
Next, we cracked the secret code as to WHEN she actually pees in her bed. This is a trial and error process, but it works- eventually. You mentioned that you tried to get her before she peed- well, keep trying earlier and earlier! You'll catch it in time and keep waking her up and taking her at that time, and THEN a couple hours after the first time. SO, do it twice for the first few nights.
We noticed that after a couple weeks of waking her up (still not dry every night yet, though), she started to wake up on her own sometimes- she had NEVER done it before!
Well, we just kept at it. It got to the point where MOST mornings the bed was dry. We got down to waking her up only once a night, and I think we just basically forced a habit with her. I think that is the key here: THEY GET USED TO US WAKING THEM UP, SO IT TURNS INTO A HABIT THAT THEY DO ON THEIR OWN!
Now, here comes the good news!!! Her bed was dry for two weeks straight! We got her her "prize" that she wanted, and since the beginning of that two weeks she has been dry since! She is also 5, and will be 6 in October. She's awesome and we are SO proud of her!
Another bit of advice- make sure you really praise her on dry mornings and make sure you don't make her feel bad on wet mornings.
Good luck, and STAY PERSISTENT! Don't give up, and keep in mind that it is very normal! I hope you find success soon!