If he hasn't paid for 8 months, honestly a formal notice might turn out to be pointless at this stage - if he hasn't put anything down for half a year, it's unlikely a formal letter will convince him. What you absolutely shouldn't skip, though, is documenting everything properly: keep your bank statements, communications, any proof showing he's not fulfilling his obligation. The court needs this to evaluate your case and for any potential penalties.
The real move is to file a claim for non-payment of child support. How long it takes depends on how backed up your local court is, but on average we're talking a few months until the first hearing, then maybe more hearings if things get complicated. In the meantime you can request interim measures to get the back payments faster, though honestly your lawyer needs to walk you through that in detail because it varies depending on the situation. What matters is moving quickly: the longer you wait, the more the debt grows, but also the court tends to take cases seriously where there's been such a prolonged total absence of payments.
Have your lawyer handle everything, don't try contacting your ex directly to "convince" him - any direct communication could make things legally messier. And tell your lawyer you don't think the formal notice will work given how long it's been: you'll decide together whether to skip it and go straight to filing the claim, or do it anyway just for the record, but at least you'll know it's not a waste of time.