To reveal their locations, go to Knockturn Alley and look at your map under the black lights there. This will reveal the spell locations written in invisible ink, and almost certainly cause you to giggle like a giant nerd for like 20 minutes. The secret spells let you make invisible ink appear on a page, cause a flower to bloom, and bring certain objects to life. In short: You're a wizard, 'arry.
That's not all: The King's Cross Station area has phone booths that look conspicuously like the one used to enter the Ministry. Most think they're just set decoration, but they do function if you know the secret number. Dial "MAGIC" (your magic password is 'magic'? That's the kinda thing an idiot would have on his luggage) and you get a special message from the Ministry. Congratulations: Your kid now thinks you're connected to a shadowy government. But, y'know, in a fun way.