Discountasp.net wants you to delegate your domain to them, I would assume they have a control panel to manage your DNS much like godaddy but you would need to check first. If they don't you would loose the ability to edit your zone file, include the MX records.
If your domain zone has been created at discountasp.net, which is normally done automatically when you sign up to a hosting provider, you can query their DNS for the records you want (A, www etc) and add them in at godaddy to achieve the same thing.
dig A +short yourdomain @ns1.discountasp.net
dig A +short www.yourdomain @ns1.discountasp.net
[user@query ~]$ dig A +short www.vps4ever.com @ns1.instradns.com
So you would create an A record with the host of www and the content of 18.104.22.168 in the godaddy control panel.
EDIT: As per your picture you DO NOT want to add two sets of name servers. Choose either the godaddy ones or the discountasp.net ones.