i would keep alex and kids apart or at least supervised, reduce your interaction with kids when alex is loose
having no idea what bird you have, this is a general kinda guild line
when birds bond they canbe extremely territorial and jealous at 1st (and how you and other interact, will determin your birds behaviour, no kissing or touching others or even standing to close, this is not for ever as over time your bird will come to accept, others in the flock, not at the same level but will not feel the need to bite etc but you may have to do this once a yr for a few months when birds hormonal)
you must learn to read your birds body language, as mentioned above, look up on the net for your bird type
also i have noticed nuts diet and excerise to be important in this changing her behaviour, pellets, fresh fruit an veg, seeds (but only a few sunflower seeds) and some healthy home cooked foods at our main family meal times, when we all eat together
she will have whats known as the parrot zooms, i leave her to it, so she can burn off the extra energy, when she flies from kitchen to room back an forth squealing a little
in time your bird will feel comfortable in its place in your family, ( and i am talking months)
this runnin up your chest and biting?? was you looking down at her/him at the time, so your face was directly over him/her or laid on your back? these use to be triggers for the nut, DO NOT be afraid of another bite!!!! if it happens think over the things that happened before bite and then try to aviod same situation, if the initial bite has made you wary, look up stick training, for your bird to step up etc
also don't let your bird get over excited (also mentioned above)
don't kiss your bird, that bite may have been its kiss back to you, and watch how you stroke it, head rubs only, as certain areas on certains birds bodies, they see as mating signals, so stroking down its back is a no no!
you will find a way to deal with these situations through trial an error, anything specific you need to know just ask
but don't feel this bird has broken anything between you's, you are its parent, so you need to guild it and teach it acceptable behaviour, and thats just time an patiance