No, what you really need to do is obtain and learn how to use modern multitrack recording software that can do what you need and much, much more.

From your desription, there should zero need for any hardware other than a computer with a 2-channel audio interface.

There are several ways of approaching this. Let's take as a given that you want to keep things in the digital domain while mixing, i.e. not convert to analog, mix and then re-digitize.

Approach 1:
A external digital mixer such as a Yamaha 01V96 would eat this work for breakfast. Mono channels can be linked as stereo pairs (vertically in layers), so you have in one layer single faders controlling 12 stereo input channels mixing down to a single stereo pair. Up to 4 stereo effects can be mixed in. What this then needs is a way of getting 12 stereo (24 mono) channels into the mixer and capturing the resulting mix as a stereo pair. One way of doing this is to use a Presonus Firestudio Lightpipe or an M-Audio Lightbridge to replay the tracks from your computer via ADAT lightpipes and to capture the result. Another way is to transfer the data files to an Alesis HD24 hard disk recorder and replay them into the mixer again via ADAT lightpipes. All this is not particularly cheap, but it gives you the genuine feel of an external mixer that you may be already comfortable with.

Approach 2:
Get a multitrack audio software program as AudioGaff suggests, and learn your way into mixing "in the box". Something low cost like Reaper would do the job and can be downloaded on a 30-day trial to see if you can adjust to this way of working.

Approach 3:
A hybrid of the two above - use a software DAW program controlled by an external hardware control surface.

Don't forget that for approaches 2 and 3 you would need high-quality stereo analog outputs from your computer, and for monitoring your mixes in all the ways of working you would need high-quality powered monitors (or passive monitors with a separate power amplifier) and an acoustically treated listening environment.

I am curious to know hat your project is overall, if that is something you could discuss on an open forum and not proprietary.
By what method were the original recordings made and digitalized?
Cool Edit Pro (which is now Adobe Audition) in its early releases was a destructive mixing program, however DAWs have come a long way and most (if not all) modern releases are non destructive editing so mixing in the box is probably the way to go. You would still require monitors and some type of interface but don't you already have some type of interface used during recording?

Anyway Boswell's excellent advice really can be looked at on a cost basis.
#1 Yamaha01v96 $2,400, Presonus Light Pipe $500.00, Monitors $400.00 and Alesis hd24 $1600 total approx. $4900
#2 Interface $300, monitors $400 total $700.00
#3 Control Surface $1200 plus #2 $700 = $1900

The #2 option's interface will likely come with a pro DAW software and seems to be the logical choice as Audiogaff stated.

Boswell wrote: Don't forget that for approaches 2 and 3 you would need high-quality stereo analog outputs from your computer, and for monitoring your mixes in all the ways of working you would need high-quality powered monitors (or passive monitors with a separate power amplifier) and an acoustically treated listening environment.

I think you would need these regardless of what method you take. Boswell's advice is good as always and far be it for me to "correct" him. His knowledge is far beyond my own.

I do think though that you should try to keep everything "in the box" if at all possible. Adding layers of A/D-D/A conversion is not going to help your project if you want to avoid adding noise.

hueseph wrote: I do think though that you should try to keep everything "in the box" if at all possible. Adding layers of A/D-D/A conversion is not going to help your project if you want to avoid adding noise.

Yes, agreed, in principle. However, I started my previous post with

Boswell wrote: Let's take as a given that you want to keep things in the digital domain while mixing, i.e. not convert to analog, mix and then re-digitize.

A digital box can be quite "stretchy" in signal terms. The point of using an external digital mixer in this context would be to achieve a way of working that uses a familiar control surface while retaining signals in the digital domain. If cost is a factor, take the DAW route - steeper learning curve, but similar results.