To get the total hours a day per client I used the following array formula:
=SUM(IF('Hours Worked'!G$4:G$32='Total Hours'!$A$3,1,0)*'Hours Worked'!D$4:D$32) Where 'Total Hours'!$A$3 refers to a specific client.
NOTE that array formula's need to be confirmed by hitting Ctrl+Shift+Enter for them to work properly.
To get the totals for the week a simple SUM formula would suffice.
Here is your file where I added the formula's in the "Total Hours" sheet for all clients for Saturday and Sunday (you can do the rest of the week) as well as the week totals:
Let me know if you got any questions or if you might get stuck.