The good news? Learning one language makes it easier to learn more. Once you know what the perfect past tense is, or imperative mood, or passive voice, or dozens of other obscure grammatical concepts we use everyday but rarely know the names for, you can apply that knowledge in other languages. So let's suppose that the implant gives you all that knowledge, then just plugs in the necessary words from the particular language's vocabulary list. Basically, it doesn't teach you a language, it teaches you language as an abstract concept. Then you download a language pack as needed.
So how big would one of these language packs have to be? After just a few minutes of research, I feel pretty confident saying that question is impossible to answer. But if we limit our language packs to just what you would need in common speech, rather than absolute fluency, we can come up with a reasonable guesstimate.
This useful source says 75% of English is made up of a thousand word bases, and 90% is made up of 7000. So we could have two levels of fluency packs, but let's focus on the smaller one. The best way I can think of to organize these packs is to list all the words in some standard order, so #1 would be be, #2 would be go, and so on. If each word is allotted 8 letters, and the alphabet is limited to 32 letters (5 bits), that's 5 bytes per word. Multiply by 1000 is 5 KB for basic usability. For the larger pack, we should allot more letters, maybe 16, and round up to 8,192 words (computer nerds will know why). 10 bytes per word times 8,192 words makes an even 80 KB. This is not much! It would also need to include plenty of information about pronunciation, conjugations, tenses, word order, and other details that make languages different. I don't have such an easy way of estimating how much space that would take up, but based on the size of the dictionary I'm guessing you could cram it into a megabyte with room to spare. This is still not much!
I've heard the capacity of the human brain cited as anywhere between three and 100 terabytes. Assuming a few gigabytes for the language base, You could easily cram thousands of languages into the brain without pushing out anything important. But of course, however it is your brain remembers English, it isn't like this. Your implants are a much better idea than trying to load the data directly into the brain - they can hold all those thousands of languages on a tiny 1GB builtin memory chip, and whenever you ask it for a particular word in a particular language, it looks in the dictionary, finds the word, applies the conjugation rules, converts it into whatever format brains use, and gives it back to you fully-formed. This completely bypasses the need for your brain to know any words at all. You use your knowledge of language in the abstract (which I'm assuming would be directly loaded, either through implants or school), and think to it, I need to know how to say the first-person singular present progressive form of to go in English, and your implant will look up go and see how to conjugate into the present progressive tense, in the first-person singular, and think back to you, I am going, and tell you how to pronounce it. Or it could do this one sentence at a time so it can handle all the grammar for you.
TL;DR: 56. (Edit: that guy's a little dubious. 68 is better documented, and higher.