It's possible that, now that you have come to the realization that you have anorexia and that something has to be done, you want to fix it by yourself too, without anybody ever finding out. And maybe you can, I don't know. In my experience however, that is not the way to do it, it doesn't work. There are several reasons for this.

First of all, wanting to do it all by yourself, not accepting help from others and not wanting to burden the people you love is part of the problem! It's part of a pattern you have developed. You are there for others, ready to help them, but you don't want to burden them, you don't want to be a problem or a hassle. Maybe you even feel like a failure if you need help from others. The thing is, disregarding your own needs is part of the anorexia. Anorexia isn't really about eating and weight, it's about who you are. Like many anorexics, you probably have low self-esteem. You don't feel like you deserve other people's attention, like you deserve their help and you don't want to ask for anything. So if you want to do this alone, you're only enforcing that, while what you need to do is change it.
Being secretive, hiding things, withdrawing into this safe world of your own, that is also part of the anorexia. You cannot recover if you don't open up to others, let them know what you are feeling and show them who you are. If you do this by yourself, if you don't allow yourself to lean on others and rely on their support, if you don't open up at all... then you're not making real, lasting changes.
If you are approaching recovery in this way, you will focus on things like eating more, weighing less often and maybe even on gaining weight. But those are just the symptoms. If you had the flu and you drank some tea to make your throat less sore, you would in fact make your throat feel better. It wouldn't hurt as much and you would feel better, but the flu wouldn't actually be better. You're just dealing with the symptoms and not with the actual virus. This is just the same. Anorexia isn't about food and weight. Those are just things you have been focusing on, just symptoms that show something is wrong. Dealing with those symptoms is very important. You can't get better if you don't start eating more and if you don't get back to a healthy weight. However, merely focusing on that won't make the illness go away. You might feel better for a while, but it either won't last or some other symptoms will evolve. Because you got anorexia for a reason.
Recovery doesn't just mean fixing the symptoms. I'm sure you could fix those on your own. Recovery is about a lot more. It's about changing patterns that you have developed, about becoming yourself again. It's about opening up to other people and to yourself. And in order to do that, you need others.
Recovery isn't something that you can rush through, something that you can start and finish quickly. Unfortunately it is a process and it takes time. Just like you didn't become anorexic overnight, you're not going to recover overnight. It would be wonderful if you could just turn the anorexia off once you decided you didn't want it anymore. But that's not the way it works. During this process of recovery, you will change a lot. A lot of things will happen and by the end of this process, you won't be the person you were. Keeping it a secret from others is really difficult. It takes up a lot of energy, energy you could really use for other things. If the people you care about don't know what you're doing, they won't understand why you are changing, and they won't make it easier for you either. Apart from the fact that recovery takes time, it's also not something that has a clear ending. There is not a single day on which you will say to yourself, there, now I'm cured. In addition to this, it will never be out of your life completely. Not because it's not possible to beat anorexia, but because the process of recovery changes you. The experiences you had changed you. You can't just ignore that. If the people closest to you never knew about this, wouldn't they be missing out on a really important part of you? Maybe sharing this is something you owe to the people who care about you.
Also, you deserve some help. If it is possible to do this by yourself –and like I said, I seriously doubt it-, it is definitely harder that way. And why make it harder for yourself than necessary? Until now, things haven't been easy. Being anorexic is no fun, is it? You haven't taken good care of yourself for a long time now, haven't given yourself what you need, haven't made it easy for yourself. That too is part of the anorexia. Going easy on yourself is something you need to learn. And you don't have to do this alone. There are people around you that care about you and want to be there for you, they can support you if you let them. I think you owe it to yourself to make life in general and recovery in particular as easy as possible.
It's also dangerous, not telling anybody what's going on. It is some sort of safety device, if you don't make it, or don't like what's happening, you can always decide to quit and nobody will be the wiser. That is a risk you don't want to take!

So, in my opinion, recovering from anorexia is not something you should (have to) to alone or keep a secret. It might be contrary to everything you're feeling, after all you've been keeping the anorexia a secret for so long. But recovery is about doing the opposite from what the anorexia tells you to. It's about standing up to it, fighting back.
It is true, that recovery is very personal. You have to do it your way, and it's about finding out who you are again, rediscovering the real you. But the fact that it's personal doesn't mean you shouldn't share it with the people closest to you.