There is no one authority for the English language.
English has no Académie française, and, even if there were such a central regulatory body, language would go its own way regardless.
In that sense, there is no answer to the exact question you ask; as one commentator on your question puts it: "I'm afraid there is no such thing as 'an official English word.' "
How to determine if a word has 'made it' into English
Fair warning: there is a large dash of my own opinion in this - apply salt liberally.
You have to look to consensus. Dictionaries are the primary way to do this so you're really going about it the right way. Look at the big names in Britain and the States (Cambridge, for one, Merriam-Webster another). See what a OneLook search turns up... and don't trust Urban Dictionary!
In all instances read the entry carefully. Just getting a 'hit' isn't enough without a thorough check and, as we're looking for consensus, one hit from one dictionary may not be enough. Consider that a word can make it into the dictionary then subsequently fade from popular usage. Also all dictionaries have their own individual biases.
Determining consensus is complicated by the reality that many disparate and diverse communities have made English their own, and brought their own baggage with them.
Is kilig a fully-fledged English word?
In this specific case, kilig is explicitly listed as South East Asian in the dictionary you refer to.
kilig adjective, SE Asian
1 (in the Philippines) causing or characterized by a feeling
of exhilaration or elation.
‘the fans went wild with the kilig moments they shared on stage’
‘it's their most kilig movie to date’
1.1 (of a person) exhilarated or elated by an exciting or romantic experience.
‘I get kilig every time I hear this song’
By inference, it's not in the 'common core' of English; the lack of results from your own dictionary trawl would tend to agree with that assessment.
Is it going to make it in? I have no idea. Is kilig being used outside the Philippines? Has it made to, say, Singapore or Hong Kong? If it hasn't got that far, it almost certainly won't fly in the UK, Canada, India or the US, for instance, nor among the many people globally who use English as a second language or lingua franca.