First, something needs to be done about this:

There were pretty often rejections or when I asked, "what are you doing today evening, I don't have anything to do today ^^"

To be honest, the way you say it is quite cringy. You sound very much like "I don't have anything better to do, so let's see each other." This is not flattering for her, as it makes her your fallback plan for when you have nothing else to do. You sound needy and bored, which is not attractive. Also, you ask her what she's doing, which makes you sound controlling, which you go on to kinda confirm:

she sometimes said stuff like: "ill be reading, or watching tv etc" which means to me, I HAVE free time, but I choose to spend it ALONE

This comes off as: you decided for her that she should spend time with you rather than reading her book. She might not agree with that attitude. I'm not surprised by her reaction, as I immediately ditch anyone who displays a tendency to think they're entitled to make decisions for me.

There are much better ways to say the exact same thing, even this old, very simple one:

"I'm going to watch [movie title], would you like to join?"

You're not bored: you are going to watch a movie. You already decided which one. Feel free to substitute with any other activity, like bowling with friends or anything else. You're asking her out because you'd like her to be there and value her company (not because you have nothing better to do). The idea is that you have a life, and you're inviting her in. You're not pressuring her to say yes; if she says no, your plans don't change. It doesn't sound controlling because it isn't.

This is what Rachel said in her answer:

Make yourself happy (love who you are and others will love you too)

If the activities involve some of your friends, this also conveys important information to her: you have friends, and you're not afraid or ashamed of your friends meeting her. It says you value her, your friends seeing you with her would be okay, and her meeting your friends is also okay, they're not unwashed creeps, they're just normal people, you think she'll like them, and you think they'll like her.

If she says "no, she will stay home and read a book", then the next day you can ask her if she enjoyed it (good point for you! you listened!) and you can talk about the movie you watched or whatever you did.

Something no-one mentioned is that she might be very introverted. Handling introverted people is a subject in and of itself, but I'll just say that they have a set amount of energy per day and interactions with people consume that energy at a rate that depends on the stakes, potential for conflict, number of people around, etc. Once the energy is drained, the introvert needs to curl up at home with a book and recharge, and especially not be bothered. If this is her case then you need to adjust accordingly and update your question.

The thing is now, that although we have a great time when we meet, and write every day, I kinda start to lose interest because everything goes to damn slow.

This is a bit like a long distance relationship then, since you only meet once a week.

Unless you are having deep philosophical discussions or other kinds of conversations that you enjoy enough to keep you interested, spending a lot of time every day writing to each other can result in boring and uninteresting small talk about each other's daily lives.

Smalltalk does not fulfill your goal of getting to know each other. So, review your conversations and try to assess the proportion of small talk versus actual content (like: talking about past life experiences, philosophy, travels, dreams, life goals, tastes, etc).

Then, rethink your interest in her. When re-reading your conversations, if you are bored into oblivion, then let's be honest, this doesn't look like it'll end up in a satisfying long-term relationship...

If however, you find her interesting, then do less small talk and explore the interesting subjects in more depth. Just don't talk about your day at all. Pick something interesting she said, and start a conversation about it.

So, to answer your question: write less often (not more). Cut it down from once a day to twice a week. This will increase the interesting discussions versus small talk ratio. Wait for her to initiate the conversation. Do not ask about what she ate, but if she cooked the recipe she learned from her grandma, then ask her all about the grandma.

Also, have her plan the next date instead of trying to control everything. If she does not, then you know she's not interested. If she plans an amazing date, then she will get you interested, which solves your problem.

In any case, NEVER say something like "I am losing interest". This is like watching a movie with a voiceover that explains in excruciating detail all the stuff that you can already see on the screen. If you are losing interest, then simply communicate/text/mail less often. If she is interested, she will then restart the conversation.