Little Witch Academia's Akko may be at an academy centred around magic, but that doesn't mean she's any good at it.
In fact, she's awful at spell-casting to be honest. In Little Witch Academia: Chamber of Time, the only way she's going to solve conundrums that require magic is by drinking a corresponding potion, which grants one free cast of a spell. Basically then, you're going to want to stock up on a range of potions until she's vastly improved her skills. But where you do you get them?
The game never really makes it clear, but there's a shop on the third floor that allows you to buy potions. Or rather, trade for them. Take a look at the in-game map and memorise the location of Academy Store B. Visit it and you'll discover that you need Spellstones to do business. You'll find them in dungeons, so go and do plenty of those to build up an ample stock of them.
Once you've got plenty of Spellstones, go back to Academy Store B on the third floor and exchange them for the magic potions that you need. Generally, they cost one or two Spellstones each. It's wise to always carry a few telepathy, movement and mending potions just in case you need them. It'll save you time, too; you won't have to keep revisiting the store as you need them.
If you're wondering, Academy Store A, south of Academy Store B, sells weapons and accessories. You don't need Spellstones to buy them though, just cash. Or Luna as it's called in Chamber of Time. There's also a save point there, so make sure you activate it so you can fast travel to it in the future.
It's strange needing potions to cast spells, but that's Little Witch Academia: Chamber of Time for you. It's a strange game, but an enjoyable one nonetheless.