My niece accidentally water my sago palm with ice cold water about few months ago. Now, there are some damages on the leaves. Should I cut it all off?
I don't think that damage is from getting watered with cold water. Most plants are not so sensitive that one brief contact with cold water will harm them.
My suspicion is that it might have spider mites. See below link for more info on how to figure out if that's the problem.
Here is a link that might be useful: Some info about spider mites with pictures
Do you think it is ok if I cut all the leaves?
Trim off the white truly dead stuff, only.