Best way to paint Ikea metal brackets?
Prep is the key

You can make it look good for 3 months with a rattle can.  If you want it to look good for 10 years, paint prep is called for.  Prep has 2 components: de-gloss(microscopically roughen) the surface so paint can bind to it, and cleaning the surface to remove contaminants.  

A 3M Scotchbrite pad, green pots and pans style, for scuffing the surface to remove gloss. Even if it doesn't seem glossy now, give it a rub.  The objective is to degloss the chrome surface, not blow through to the aluminum layer.  

Common bathroom alcohol e.g. Isopropyl, with a double cloth wipe, to remove contaminants.  

Rattle-can spray primer such as Rustoleum 7779 rusty metal primer.  Take the time to learn to apply paint competently.  

Nickel and aluminum are a challenging surface for Rusto metal primer, but I gather a mil-spec aluminum primer (the green stuff on unfinished airplanes) is out of the budget.  (It doesn't come in cans because it's 2-part; you mix the 2 parts minutes before you apply it.  Also, the B-part is toxic until it cures, so, not a thing they want in the consumer marketplace. 

Rattle-can spray paint for your paint layer.