The psionic power mindlink does exactly what you want: It's low level and works on any creature that has an Intelligence of 3 or higher. (As long as you spend the points to make it work on an unwilling target.)

And most importantly:

You can communicate telepathically through the bond even if you do not share a common language.

As a backup option, as long as you can touch the ice toad, you can cast tongues on it. Tongues doesn't require a willing creature, so as long as the ice toad fails its will save this should allow it to speak to and understand you.

This spell grants the creature touched the ability to speak and understand the language of any intelligent creature, whether it is a racial tongue or a regional dialect. The subject can speak only one language at a time, although it may be able to understand several languages.

This won't allow it to speak to other ice toads:

Tongues does not enable the subject to speak with creatures who don’t speak.

So if you want it to be able to speak to other ice toads, you'll have to cast the spell on them too.

There is one other fly in this ointment:

The subject can make itself understood as far as its voice carries.

This is potentially problematic, because if ice toads don't have a voice (which may or may not be the case), they will be able to understand you but not speak loud enough for you to hear. This will enable any communication method that relies on the creature understanding you, and might allow communication methods that rely on the creature having a language. (Depending on whether understanding every language counts as "having a language"; I can't find a rules definition of what that requirement actually entails.)

Alternatively, with one-way communication established, you may be able to talk it into accepting a telepathic bond or some other method that relies on having a willing target.

No

Player's Handbook page 7:

If a specific rule contradicts a general rule, the specific rule wins.

... For example, many adventurers don't have proficiency in longbows, but every wood elf does because of its racial trait.

This can easily be extended to the question of Kenku as such:

Most Storm Sorcery Sorcerers can speak Primordial, but if he/she's a Kenku, he/she cannot because of its languages.

...but you can speak only by using your Mimicry trait.

You would, however, now be able to understand Ignan, Aquan, and Terran and, after conversing with creatures who speak these languages enough, pick up enough phrases to use Mimicry for these languages too.

Why is the Kenku language more specific than the Storm Sorcery feature?

To identify which rule is more specific, you have to identify which rule is general. In this case, the general rule is Wind Speaker, because it can apply generally to all races while Kenku language restriction only applies to the Kenku.

If Kenku was the general rule, then Wind Speaker would have to clearly allow speech beyond the Kenku restriction. It is, however, clear that Kenku can not speak without Mimicry,

specific: (adj.) clearly defined or identified.

Similarly, the specific case of being mute would also restrict you from speaking with Wind Speaker (since it doesn't specify speech beyond normal means).