I know this thread is slightly old now but I am desperately struggling with the same issue. My host is Crazy Domains who provide no access to the MX Records.
When you say, "delete the existing MX Record for the domain and then add 2 new ones; the zero and the 10 and get the values for each from your client", do I get these values from the people who are hosting the emails? And what should they look like? An IP address or like a nameserver?
Any help would be greatly appreciated!