It's leggy and slow-growing because it needs more light. Move it closer to the light source (which you say you've already done) and she'll improve. In the meantime, train a little fan on her - it won't reverse the stretch, but it will strengthen the weak stem and stop her from falling over.
RE: nutrients: the seed itself contains all the stuff the plants need for the first week or two - once you've got two sets of leaves you can start with some nutrients. You're growing in soil, so when you need to start to adding nutrients will depend on what soil you're using...
I can't see from the grow diary what you're using. If it's a pre-fertilised soil (e.g. Biobizz all mix, Canna Professional Plus, Plant Magic, etc.) then you can probably hold off for another week. If it's not (e.g. Biobizz light mix), then you can start adding some nutrients now. Just start off at half the suggested dosage. You can increase the dosage as the plant gets larger and once you're confident she's taking it down. Water until you get about 10% run-off and alternate between plain water and nutrient solution every feeding to prevent build up salts and waste nutrients in the pot.
Good luck! :)