I had a similar problem recently. Except laptop was fine but my Kindle was being rejected after months of no problem. I did the usual fixes but nothing helped until I replaced my router. No problems since.
It's not 'handshaking'. I don't know what make it is, but if you google '*make and model* of notebook' not connecting to internet, or 'make and model of notebook' not handshaking, hopefully there will be a selection of answers for you.