Unknown Litium Ion Batterys
Dimensions and type:

Cylindrical cells with types like 14500 and 18650 use a naming system of
diameter in mm + length in mm + 0
So


14500 = 14 + 50 + 0 = 14mm dia, 50mm long    
18650 = 18 + 65 + 0 = 18mm dia, 65mm long  


Diameter dimensions are reasonably close - say +/- 0.5mm usually.
Length dimensions may be a few mm different.
Nipple ended and/or "Protected" 18650 cells tend to be a few mm longer than flat ended and/or unprotected cells. 

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Cell &amp; battery Voltage, mAh capacity, Wh capacity

LiIon cells have usual operating ranges of 2.5V - 4.2V but discharge below 3V gives very little energy and risks cell damage.  Cells under 2.5V may be dead. Cells under 2.0V are probably dangerously dead. A cell in the 2.0 - 2.5V range may be test charged at VERY LOW current - say C/100 or perhaps &lt;= C/10 to see if it charges up to > 2.5V. Charging a cell that is below 2.5V at full rate risks catastrophic failure. Charging a  cell that is below 2V at high rate is an act of lunacy.

Charge rates for good cells may be as high as C/1 but C/2 is safer.

Cells are considered to have nominal (or average) voltages of 3.6V or 3.7V when batteruy voltage is calculated. So your 11.1V nominal voltage indicates 3 cells in series (3S) as 11.1/3 = 3.7V per cell.

Cell Wh capacity = nominal voltage x Ah capacity.
Ah = Wh/V
53 Wh/3.7V = 14.32 Ah at 3.7V.  .
If there were 3 cells Ah/cell = 14.32/3 = 4.77 Ah/cell.
Too high.
If there are 6 cells then Ah/cell = 14.32/6 = 2.39 = 2.4 AH/cell.
This is about right for cells produced some years ago.  

So the battery is arranged as 6 cells in 3S2P configuration with 2.4 Ah/ cell.  

Max charge for good cells at C/2 = 2.4/2 = 1.2A max charge per cell.

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Suggested procedure:

Be aware that there is a potential for 'vent with flames' events or (less likely) possibly more dangerous self dismantlement. 

Separate cells into 3 groups   

V >=3V
2.5V &lt;= V &lt; 3V
V &lt; 2.5V  

Cells &lt; 2.5V are probably dead, but maybe not. 

Use a charging source of > 3.9V and =&lt; 4V.
NOT 4.1V. NOT 4.2V
Cells may be 'floated' semi-indefinitely at &lt;= 4.0V.
Doing this at much above 4V risks permanent cell damage or catastrophic failure. 

Low or very low V cells have a low "trickle up" charging current applied to see if they can be charged back into the normal range. Cells that have been below 2V are very likely badly damaged or dead. Force feeding a cell measuring &lt; 2V may cause catastrophic failure. 

Assume trickle up current of &lt;= ~~ C/50 = 2400 mA/50 ~= 50 mA
For &lt; 2.5V cells charge cells via series 47 to 100 Ohm resistor.
( R = V/I = 4V/0.050A = 80 Ohm)

For &lt;= 2.5 cells charge from 4V via series 4.7 to 10 Ohm resistor.
(ie at say ~~= C/10. I = V/R = (4-2.5)/0.24 ~= 6 Ohms.)

If cells under 2.5V fail to rise to >=3V after 'some while' discard them in an appropriate manner.

If cells > 2.5V do not continue to charge towards 4V discard them.