Summary: The cure to hard or overly chewy fat is generally longer cooking, but you may also want to consider trying a different source for procuring steaks.
My guess, based on the description in the question, is that this is a collagen issue. Beef fat will begin to melt over around 100F (~40C), and it will begin rendering in earnest by the time you hit the "medium rare" temperature. (That's often why "medium rare" is considered ideal for steaks; it's not as tender as rare or "blue" meat, but the fat rendering makes it taste more succulent.) If the steak is really done to "medium well," it should be plenty warm enough to get the process of internal fat melting going well.
[Sidenote: I'm not sure I understand the question description of the fat as "cold" or "uncooked" -- it's basically impossible for internal fat to have those characteristics when surrounded by muscle cooked to medium-well, unless you have a giant vein of connective tissue running through the middle of the steak. However, fat has a different heat capacity than muscle, and it often seems to be at a different temperature when eaten. I'm guessing that's what's going on here?]
So if the fat is still tough, the problem seems most likely to be collagen content. Collagen gradually converts to gelatin during cooking, a process that speeds up as the cooking temperature gets higher. While collagen is found in greater quantities in tougher muscles (e.g., the chunk or shoulder), it also helps to hold the structure together even in more tender cuts. Often, the stuff we call "gristle" in fatty steaks isn't the true gristle of elastin and tendon which wouldn't be broken down with prolonged cooking, but merely fat or connective tissue with a higher collagen content. In a pot roast cooked for many hours, this stuff would gradually turn into gelatin.
Anyhow, there are many different types of collagen too, some of which take much longer or higher temperatures to break down. Depending on the diet of the animal, how it is exercised, how old it is at slaughter, and whether (and how) the meat is aged, the collagen around and within some sections of fat -- even typically "tender" fatty cuts like ribeye -- can be quite tough and resistant to breakdown. The biggest factor is often age: as an animal gets older, the types of collagen networks get more complex and harder to break down.
If this is indeed the issue in your steaks, there are two solutions:
Try buying different steaks. It could be your source. I don't know where the OP is from, but in the U.S., grocery stores (for example) often sell ungraded meat under "store brands." The meat isn't necessarily bad quality, but one reason to not bother to grade the meat is that the animal is a little older at slaughter, which automatically will disqualify the meat from the higher grades most people look for. Or it could contain very little marbling, which tends to mean that the fat which is present has a higher concentration of connective tissue. I've had plenty of grocery-store steaks over the years which had flavorful muscle, but most of the fat would never render and would remain tough, likely due to higher collagen content. This seemed to be much more common in steaks from some stores, but not at others. (Also, if you're in the U.S., be careful to note that graded steaks should specify USDA grading and generally carry the USDA seal -- it's actually perfectly legal for stores to sell an ungraded steak under a label like "America's Choice," even if the meat is not USDA Choice grade.)
If you can't change sources, or the problem still occurs, the solution is longer, slower cooking. Perhaps a lot longer. Collagen breakdown can occur at lower temperatures, but the higher the temperature, the faster it will happen. If you want to cook for a long time but still maintain a medium or medium-rare steak, the ideal method is sous vide (i.e., sealed meat in a water bath held to constant temperature). Alternatively, I have had success with a long, slow-roasting method cooking a steak in a very low oven (perhaps one that is barely heated to "warm" and then turned off; you'll probably need to monitor the internal steak temperature with a probe) for a couple hours, then searing very quickly before eating. This latter method often gets me closest to the "steakhouse" experience at home, since the slow-roasting at low temp effectively does a bit of accelerated dry-aging, and the fat in a cut like a ribeye will often end up soft, buttery, and ready to burst with juices. (However, even after a couple hours of slow cooking, I've still had some grocery store steaks where the fat remained tough. I stopped buying steaks at those stores.)
Just as another note: I mentioned above that I didn't really believe the fat could still be uncooked and cold in a medium-well steak. If, for some reason, I'm wrong, and that is actually happening here, the solution will still be longer, slower cooking to even out the temperature as the meat cooks. You may not need to go to the extremes I mentioned in the previous point, but you'll perhaps want to slow down the cooking process significantly.
Of course, these alternate cooking methods will take long, and most people want to cook their steak for a few minutes and then eat. In that case, I'd strongly recommend trying meat from a different source and see if your results improve.