Add DKIM So Can Send Email From Other Domains

I don't know exactly what to ask so I will explain what I'm trying to do instead.

I have 2 domains and 2 different VPS. I want to allowed domain1 to send emails for domain2. I want able to do it with SPF like this. On domain2 dns added "domain1.com. TXT v=spf1 a mx include:domain1.com ~all". That works.

I am trying to achieve this with DKIM like this.

On domain2 dns added "default._domainkey    TXT    v=DKIM1; k=rsa; p=MIGfMA0GCSqG................................... " but that didn't work. When send from domain1 for domain2 emails the header said dkim=none and Received-SPF: Pass according to hotmail.

I don't know what I'm doing so that could be the issue here. I just google and read things and try it as best. Can someone please help? What is the correct way of doing this?