Is a 6x6 beam good enough for a span of 10ft
No. 6-by lumber hasn't been used for beams since the 60s or so (and it doesn't fit in your wall anyway). An engineer would specify something between a doubled 2x10 and a doubled 1-3/4" LVL (engineered lumber) beam, up to 12" nominal height, depending on the roof system. 

If I was held to the fire for a guess I'd put in a doubled 2x12 in a U configuration with a flat 2x4 on the bottom. The primary factor is whether you have a full-span engineered truss system or a conventional ceiling joist configuration where they'll be carried by the beam.

Simply put, we can't say for sure from our desks what you need, but I can say that a 6x6 isn't it. They're for posts, not beams.

Also be aware that it's not uncommon for beams that long to have twin trimmers at each end. You may also need to block through the floor system to the wall or beam below as well, to prevent overstressing the subfloor at those concentrated load points.



UPDATE:

From the new photo it looks like that wall was open at one point already. The key question is whether the opening spanned the entire width of the header. Sometimes single headers are used with intermediate trimmers acting as posts. 

I'd still remove everything up to the double top wall plates and install a double 2x12 header immediately below that, replacing the 2x6 header and resting on the existing trimmers.

Disclaimer: I'm making a number of assumptions here that aren't supported by adequate knowledge of your building. You should be consulting a local engineer, not a bunch of strangers on the interwebs. Take my advice solely as a presumed plan of action to be verified before completion.