First, as you may know, the Slime Eyes must first be found by slamming into a wall with the Pegasus Boots. Once that happens, you have to smack it in the middle until it is almost divided in two, then finish off with a Pegasus Boots charge. You'll know when it's thin enough when it seems to no longer react no matter how much you hit it. That will split it into two, and that's when you can start truly damaging the boss.
I recommend equipping yourself with the Roc's Feather in addition to the sword now. Each time you strike one of the two halves of the boss, it will leap up, and then slam into the ground at another spot. Unless you jump, you'll get stunned when it lands, so that's half the reason why you have the Roc's Feather equipped. The other half is because it can offer you some safer mobility to dodge or attack the Slime Eyes (jump attacks are very potent for extending your range if you play your movements right)
Try hitting them both at the same time with a Whirlwind Attack. Not only does this do substantial damage, but then both should rise mostly at the same time. This gives you a lot more breathing room than if you alternatingly strike them, or if you try to focus down one. Note that after hitting one, try to get close enough to strike one or two more times with normal slashes. There's a small window, but a window nonetheless, where you can sneak in additional hits before it flies up. They don't actually have a lot of health at this point, so it may only take as few as 2 rounds to vanquish the Slime Eyes.
The main focus is on keeping the pressure off of you. As long as you make sure to jump when they land, and also avoid being smashed as they land, then the push back from your sword should always keep them at bay. Your best defense once they're on the ground is a strong offense, so don't concern yourself too much about dodging. If they get close, swing!
If you still have trouble, by this point you should be able to purchase Secret Medicine from Crazy Tracy, located in Koholint Prairie that is south of Tal Tal Heights. This will heal you to full hearts should you lose them all.