how to drive 5x5 IR LEDs with 100 mA, using Arduino
First, a safety warning

25 IR LEDs at 100 mA each will be a LOT of IR radiation.  If they're close together, your pupils are dilated (indoors, this is often true), and/or you're close to the matrix, you could really hurt your eyes.  

The image of the matrix will be focused to a tiny area on your retina, and you'll hear/feel a little pop as the blood and fluid there boils.  You'll have a permanent blind spot.  Not fun.  BTW, it's your job to make sure that you understand what you're doing and don't hurt yourself, not mine.  I'll help you begin understanding, but I won't be held responsible.

My advice:  For development (and production if possible), put bright green LEDs in series and physically close to the IR LEDs so that your blink reflex is activated at a bare minimum.  Use hardware (in addition to software) techniques to limit the number of LEDs lit and/or the power delivered to the matrix to help ensure that only one LED is on at a time.  

Techniques

There are many LED driving techniques.  None of them hinge on delivering 1.35V; that will change between LED batches, over time, and with temperature.  

If you're just interested in current limiting, a few transistors will be sufficient.  The total transistor count will depend on your choice of a current limiting circuit topology, which is dependent on your heat sinking capabilities, routing area, and other contstraints.  There are also voltage/current regulation ICs and linear LED drivers which would simplify your job.  If you're interested in power conservation, many buck converters can be configured for current limiting, which would maximize your battery life.

You may want to have unlimited sink and limited source (or vice versa).  Alternatively, you might have just one current source, and mux it between the various LEDs.  Because of the safety issues inherent in this project, I'd recommend the latter option.