Pushing the conversion rate from signup to paid closer to 100% means crippling the free plan, probably killing the chance of viral growth.
There's definitely room for optimising throughout the funnel though.
Also, I could look into raising the price, offering additional features, etc. An interesting source of revenue could be selling books via the site, although I have a feeling that dealing with publishers would be difficult.
That's pretty reasonable. Of course pushing the numbers up means fewer people will sign up at all (i.e. not every signup can be converted), so there'll be a trade-off of some kind.
At any rate, it might be worth setting a top-end of what kind of growth you might be able to expect and try and build your business model out that way. Right now the number of uniques you need to come seems a bit high to me. Might be worth modelling out what you could do with 1 million uniques, 10 million, etc. and try and model around that, then go out and try and get those eyeballs to your app. That kind of analysis can help you decide on marketing approaches. 10 million, for example, might require some investment in actual advertising, radio ads, late night tv etc. 1 million might be easier to do for "free" + your time.
Some things that might help get more eyeballs on your site
Once you're satisfied with where the site/product is, you might think about taking a few days and writing to sites like LifeHacker as well as language learning sites, any reddit about languages (e.g. /r/korean) to help kick things off (see below)
Other than time, these can be decent free ways to kick off some growth as well.
note: I've gotten some decent notice through the following in the past just by writing them with some information. Eventually most were able to carry it, and often it spilled over into twitter and drove traffic for a while while the news bounced around the globe. It didn't cost anything and was entirely free.
Also might get coverage in various newspapers tech sections, global news or travel sections.