According to the manual ( http://www.allen-heath.com/media/AP6822_5+ZED14_18_24UG.pdf ), the line inputs on this mixer are balanced. So, use an XLR to TRS adapter cable and that takes care of the balancing issue.
However, the bigger problem here is the level of the signal you are dealing with. The line input channels have a max nominal sensitivity of -26dBu while the mic inputs have a sensitivity of -63dBu.. so you cannot plug any microphones into these inputs without a preamp. This will be a problem for your guitars.
For bass, check to see whether you have a line level output on the amp being used.. some amps don't have them so check ahead of time.
For keyboards, if an amp is not being used, you will probably be ok taking the output of the keyboard.. either directly or via a balancing box. In this situation you actually don't want a regular DI because the impedance conversion (high impedance line input to low impedance mic output) will result in a lower level coming into the mixer, which is the opposite of what you want. Ideally you want something with 1:1 isolation such as the ART DTI which gives you two channels of 1:1 balancing in one fairly inexpensive box.
Of course if there's no amp for keys, you need to provide a monitor channel for them.
If an amp is in use and there's a speaker output that is active at the same time as the internal speaker, there are DI's with attenuators suitable for connecting speaker outputs to mic or line inputs.
Bottom line is, make sure you know what sources you've got to work with going into this. If you need two more mic inputs for the guitars anyway, you might want to just borrow or rent yourself a second small mixer somewhere to submix the extra inputs.. for example if you submix the drums on another mixer and bring that into a line input you'll have enough mic inputs for everything and will be able to avoid the line input issue, then you can mic your guitar amps without issue and use ordinary DI's for the line sources.