IC for Trickle Charging vs. Fast Charging a NiMH rechargeable battery using solar and wall charger?
Well I am not sure how much my answer will help you, but I am working on similar project of NiMH batery charging for last 2 months and putting what I have learned till date.


  
    My confusion is about trickle charging the battery vs. fast charging it. I know that the trickle charging rate for a NiMH is between C/10 and C/40, which comes out to 245 mA and 61 mA, respectively, and fast charging is between C/2 and 1.2C
  


Trickle charge, as you said C/10, C/40 is usually very low charging current and is good for long battery life and mainly used in applications where you are not expected to the replace battery for months.

Fast charge is can be between C, C/2, 2C also and is used to revive the batteries quickly and come into action, but with this charging the battery life is reduced.

You can look into this link for specifics. So it depends on your application what sort charging you should go for. Tricke or Fast. And there is another term also called as Top-Off charge which is basically a maintenance charging.


  
    My problem is finding an IC that can do both, specifically for NiMH batteries.
  


For fast charging we tested TI part BQ2002 which will only do fast charging and is good for 5V wall supply.Here is a 12V application circuit using BQ2002. 

Then comes the trickle charging we tested BQ24400 and this IC controls the charging current based on the Rsense resistor and you can manually set the charging current based on your requirement.

I understand you are actually looking for a combination or merged properties of both the ICs I suggested, but if you are okay to go with some charging current between Fast and Trickle charging current, BQ2002 may be useful for you.

Now another important point is to take care for both the ICs is there charge qualification stage, they actually scan the battery voltage and temperature regularly before deciding which charging need to go. And when to terminate the charge, will again depend on the battery voltage and temperature of the  battery. If you are not willing to use the thermistor for temperature sensing(like me) make sure to fool the ICs by providing the expected voltage on the Tsense pins to qualify for the charging. We have tested the fooling option it works fine.

And do not misunderstand me as TI agent, due to availability of free quick samples, we tested their parts as of now.