choXer's answer already covered most if it, but it seems like you need more detail. First, I can see why your question got a downvote, though I'm not convinced it was justified. We get a lot of questions like "Please translate this," or "Please check my grammar on that." One general criterion for a "good" question here is that there might be other people who have a similar question and the answer will be helpful to them as well. Translation and proofreading do not meet that criterion so requests for them are usually downvoted and closed fairly quickly. It may help to rephrase your question, especially the title, to make it sound less like a proofreading request and more like a question on grammar, which, if you read between the lines, I think it is.
There are two grammatical errors in your version. First gerne is not an adjective but an adverb, meaning it modifies a verb and not an noun. Adjectives that mean "happy" in German include froh, zufrieden, glücklich, and erfreut; which one is most appropriate in this context is a topic for another question. These can all be used to describe yourself (ich), but gerne describes the action, and can be translated as "with pleasure" or "happily". So the sentence Ich bin gerne. would mean something like "I exist happily," which may be true but one generally doesn't feel the need to communicate it.
In general, I think German is more likely to use the verb itself rather than to use helping verbs than English is. So you should try to phrase a sentence with the main verb in the main position rather than delegating a helping verb to do the job. The English, "I am happy to write ..." puts "to be" (in this case "am") in the main position, but in German you would normally put the main verb schreiben in the main position, so Ich schreibe .... Besides, when schreiben is in the main position then gerne is applied to it rather than sein, so Ich schreibe gerne. - "I'm writing with pleasure." is closer to your intended meaning. You could rephrase it with sein (see choXer's answer), but I think the result would sound unnecessarily wordy and awkward, something like "It is with pleasure that I write you in German," sounds in English.
The second error is that that you had two conjugated verbs (bin and schreibe in the same clause, which is normally not allowed. You'd need a second clause to carry this off in German, but then, as mentioned above, the result would be more wordy than necessary since the thought is simple enough to express in a single clause.
I would translate your example as Ich schreibe Ihnen gerne auf Deutsch. This is the normal word order: subject + verb + other elements. Ihnen, since it's a pronoun, falls soon after the verb, and auf Deutsch, since it's the new and interesting piece of information, falls last. German word order is more flexible than English, so there is some room for interpretation on what the exact word order should be. In particular, the Google version moves gerne before the verb (moving ich after the verb to preserve the V2 rule), which basically makes the sentence about what you are happy to do. This is grammatically correct but I don't think it's really the emphasis that was intended.