Hello
Obviously deleting the first site is not the one that causes the problem. But cleaning the second site might be not enough. It really depends how many domains you have on the same shared hosting account. If you have many more, basically every one of them should be checked, otherwise you risk to get the malware back in no time.
So far google reports: Malicious software includes 2 scripting exploit(s).
I would need ftp access to the site, download the files and than check for the bad script and remove it. The other problem you face is the following: even if the malware is removed, google have to be contacted trough google webmaster tools for re-inclusion. So basically removing the bad parts is not enough to get your site back.
The third problem is that without implementing methods to avoid future infections, basically you are constantly at risk to get infected again.
While the removing of the script from one site can be indeed a 15-20£ job, searching trough on more websites on the same shared host is cannot be done for the same price. Also if there is research needed to find where the infection can get in, and make measures to avoid it for the future, again is not a 5 minute job.
Anyway, feel free to hit me up for a chat to tell me more about your host/website.
Regards
Eva