日本語で話しかけるね means he/she confirms that he/she will speak to you (words at least like こんにちは, こんばんは or 元気ですか at the beginning) in Japanese.
You are asking the tense of ‘日本語で話しかけるね’, right?
The sentence describes his/her will to speak to you in Japanese, in other words, he/she is supposed to do so (in the future). So that it is categorized in ‘non-past’ in Japanese language. Because Japanese language doesn’t have ‘future tense’, but categorizes two ways which are ‘non-past’ and ‘past’, by conjugations as follows:
話しかけます(formal) / 話しかける(informal) → non-past
話しかけました(formal) / 話しかけた (informal) → past
Specify the tense is one of big issues and the experts have been working on how to translate it to foreigners. We learn Japanese in the different way to that foreigners do. So the above category comes from Japanese grammar books written in English for foreigners.
People recognize the sentence if it is for future out of ‘non-past’ in the context. When we are talking about the future, speakers use the word such as 次回 / この次 (next time), これから(from now on), 明日(tomorrow), etc.
So, I suppose there must be something like one of those in the sentence he/she wrote to you.
Regarding the meaning of ‘hanashikakeru’, in your case, it is rather the latter out of Timml’s explanation. Throughout the meaning, ‘hanashikakeru’ is more focus on just before and after starting conversation than ‘speak’ 話す(hanasu). It also, indicates clearly who is the first person to speak, especially in the past.
Here is the examples of 話しかける in past:
If you say,
町で友達を見かけたので 話しかけた。It means you saw your friend in the town, and you addressed him/her. (You were the first to speak)
町で友達に 話しかけられた。(passive) It means you were addressed by your friend in the town (Your friend was the first to speak)
ねis indicating his/her confirmation in a friendly way.
◆ I added (words at least like こんにちは, こんばんは or 元気ですか at the beginning) in the first line.