I don't know your mixer, but I greatly suspect you cannot send to tape out at the same time as monitoring the tape in. Selecting tape in usually deselects anything in the rest of the mixer - might not, but since you don't hear anything...
Do you have a monitor speaker/amp set up, or are you relying on headphones in the mixer?
I think the heart of the problem is the facilities of your mixer. There are several ways it can be done with any mixer, but ideally, a small recording mixer has an extra stereo send selectable from any input channel. Such mixers either have an extra out bus called ALT or SUB or ControlRoom, which feeds the 2 ins of the sound card. Lacking this, you can use pre-fader AUX sends - but many mixers only have one prefader AUX.
Actually, Spirit have finally done the obvious and put little red RECORD send channel buttons on a new range of budget mixers, but that doesn't help now
Never mind CoolEdit for now, I sugest you try a little test...
Plug a mic into your mixer, turn it's channel fader down, input gain up and the AUX send up. Cable the AUX out to any of the cards inputs (check this with the cards manual, it's easy to mixup the in and out RCAs. If the mic is a Condenser - you may have to switch Phanton Power (48volt) onto its channel.
Open the Delta mixer. You should definately see the 1/2 input meter responding as you talk into the mic, as well as the Master output. With the Patchbay setting you have (monitor mix to HW 1/2) , connect both card outs to a stereo input of the mixer.
With this stereo channels fader turned up, you should now be able hear the microphone in your headphones coming back from the soundcard. If that works, try it again using the Tape In.
Note - the mixers PFL buttons effectively (usually - I don't know your model) Solo channels to the headphone monitor. So watch for that when using the Phones.
When you prove the link between mixer and card CAN work, then it's time to try recording something. Well actually, the next step is to see if you can hear a sound file playing from CoolEdit.