Eight lines, two verbs doing all the work. - I have a doll - is tener in its plainest job, and in the second verse shows the same verb picking up an object pronoun. If you can sing this, you can build half the sentences a beginner needs.
The gift for an English speaker is me constipó. It does not mean what you fear: constiparse is to catch a cold. The doll has caught a chill, which is why she is in bed "con mucho dolor". File it with the great Spanish false friends and enjoy correcting people at parties.
Two wardrobe words come glossed rather than drilled: a is a little chemise (the diminutive -ita softening camisa, shirt) and a canesú is the yoke of a dress - a French loan that shows how much 19th-century fashion Spanish borrowed from Paris. The 1895 collector caught the first two verses; the doctor's visit and the counting rhyme grew on afterwards, the way children's songs do, and are sung wherever the song is sung. The prescription - syrup, administered by fork - is the best absurdity in the Spanish songbook, and the counting coda turns the tune into your first arithmetic drill: dos y dos son cuatro, all the way to dieciséis - then the proud last couplet: times tables learned, homework done, vamos a jugar.