How to make a balanced battery pack?
Since this is for my workplace I do not want to use LiPo because they are so volatile. 


Lithium Ion cells are MORE susceptible to "vent with flame" / "magic smoke" events than LiPo (Lithium Polymer), not less. 

If you want a much safer Lithium based chemistry cell then use LiFePO4.
 (LiFePO4 = Lithium Ferro Phosphate).
 You can drive a stake though their hearts (metal or not, at cross roads and at midnight optional) and they will not produce high energy discharges. They will (probably) not work very well thereafter




  I would like to make a lithium pack for this since 3 lithium cells would be perfect. ... Can someone please tell me how to do this? Also, can I use a LiPo charger for Li-Ion or do I need to make my own charger?


LiPo and LiIon have similar charging needs.
 LiPo usually allows larger charge rates but is essentially the same chemistry.
 A charger must "respect" the maximum battery  voltage and maximum charge current - these are usually selectable. The per cell termination voltage for LiIon or LiPo is usually 4.2V at 25 degrees C. Using a slightly lower voltage than this reduces capacity but greatly enhances cell cycle life. 

A LiPo or Lithium Ion cell provides from 3.0 to 4.2 V/cell (discharged to fully charged) and cells are rated at 3.6V or 3.7V nominal
A LiFePO4 cells provides about 2.8 to 3.6V cell and are rated at 3.2 or 3.3 V nominal. (Lower is possible but there is little capacity in the "tail" and using it reduces cycle life). 
 A NimH cell produces about 1.0 to 1.3 V/cell and is rated at 1.2 V nominal

So your  10.6V NimH pack = 10.6/1.2 = 9 cells   

Across charge range:  


9 x NimH = 9V - 11.7V , 10.8V mean
3 x LiIon = 9V - 12.6V  , 10.8V mean  
3 x LiFePO4 = 8.4V - 10.8V, 9.6V mean   
Lead acid = 12+ V mean


You can stop discharging the LiFePo4 cells at somewhat higher than 2.8V so that the range is say 9V - 10.8V

You will need to determine how low a voltage the servos will run on happily but the LiFePO4 cells will probably be acceptable and are far better for safety. That said, properly run LiPo cause reasonably few problems.
 Using 4 x LiFePO4 cells could be considered. 

You may wish to consider the use of a 12V lead acid battery. These are cheap, a wide range of capacities are available and chargers are readily available. 

You could use a boost converter from a lower voltage or a buck converter from a higher voltage to get good constant voltage operation.



Balanced charging

LiPo cells can be charged using readily available "balancing chargers" or charged separately in parallel or you can make your won charger. A range of balancing charger ICs are available from eg Digikey.
 A simple method to ensure balancing is to place a clamp regulator across each cell which shunts current when voltage reaches the constant voltage charge point. There are various ways of doing this but one such is shown below. This is from here.   Good notes on operation and adjustment are shown on that page.
 The IC monitors battery voltage and when a trigger voltage is reached the MOSFET is switched on to shunt current. The IC contains an internal voltage reference and provides programmable hysteresis for the clamp voltage. Use of one of these circuits across each cell allows use of a non balancing charger up to the current capacity of the clamp.  

Black belt holders only In fact, although this would not be recommended, you could probably use a simple 12V lead acid charger with a current output no higher than the capability of the clamp circuits to charge 3 x LiIon or LiPo cells ion series. This would not provide some of the finer points of LiIon / LiPo charging but should work. 

MAX921 datasheet. These are far too expensive at $4.60/1 Digikey.
 Same result can be had with a TL431 and maybe a few transistors. A workable circuit can be arrived at if these is enough interest.