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Remember that judging the flare, is a learned skill, just like the other skills you have learned in your first 20 hours. Once you’ve learned how to do it consistently, you don’t even think about it. I remember when I first started teaching in gliders, I had trouble breaking down the skills and judgment necessary.
Good advice from scubawasp. Use full flap, then you can reduce your approach speed. There are hardly ever circumstances where you need to use less than full flap for landing. Even with 20° of flap, 60 kts sounds fast to me. Check the manual for the flaps-down stall speed (Vso) of the particular 172 you are flying and then multiply by 1.3 to get a full-flap approach speed.
I’ve seen Vso of 43 kts on the Internet, which gives an approach speed of 56 kts. Bear in mind that Vso in the manual is for operations at gross weight. Vso at typical training weights will be slightly slower. Your calculated speed may need to be adjusted in gusty conditions. You should discuss this with your instructor.
As an aside, I’ve noticed when being checked out in various places, that training organizations or even individual instructors often add a few knots to the approach speed “for safety”. I think this is a bad practice and often leads to new pilots having difficulties landing and can contribute to “wheel barrow” landings and occasionally nose-wheel failure.
As far as judging the flare, try getting your instructor to fly you down the runway at flare height, to get the “sight picture”. Once you start your flare, deliberately transfer your gaze from your aiming point to the end of the runway. If you notice that you’ve left the flare too late, you’ll need a more aggressive but still smooth pull on the yoke. If you have flared too high, a smooth and relatively small increase in RPM will help reduce your rate of descent.