Iglishmêk is essentially a sign language, perhaps much like American Sign Language or the signals used to communicate in professional baseball. The Dwarves had many different varieties, which were quite elaborate. Iglishmêk was used "for secrecy and the exclusion of strangers", so it's likely the Dwarves would use this system of communication at the same time they were speaking. One could envision a scenario where there were two conversations taking place concurrently: one where the Dwarves spoke with other races, and one where the Dwarves signed among themselves to discuss the spoken dialogue.
In The War of the Jewels, iglishmêk is referred to several times as 'gesture-language' or 'gesture-code', and "gestures" are mentioned often to describe it. It therefore seem natural to start at that point to determine an analysis of the word. Also, the word aglâb is discussed in the same section, and means "a spoken language". Since both words start with apparent roots of '-G-L. The initial reaction would be to assume that '-G-L refers to "language". However, it is then difficult to account for the difference in vowels: 'agl- versus 'igl-. Looking at the words for "language" or "dialect" in Arabic, one finds that they are derived from roots that relate to either "the physical tongue" or "to speak". Since iglishmêk isn't associated with the tongue, we can probably rule that out. The roots for "to speak", of which there are several in Arabic, all produce words for "speaking, speech", "word", "utterance", "saying", etc. This becomes interesting because "word, utterance, etc." can lead to a derivation for aglâb, as described there, and we can use "speaking, speech" in iglishmêk instead of "language" or "code".
We can then approach iglishmêk from the view that it means "speech of gestures", which seems quite appropriate. "Speech" would be an example of what Arabic calls a "masdar", which is a gerund or verbal noun. As an example, the English verb "to run" would have a gerund "running", which is a noun meaning "the act of running". Looking at the forms that Arabic masdars can take, it turns out that CiCC is one of them, which we see in 'igl. Unfortunately, Arabic gerunds are formed irregularly from their base verb, so we can't use that to guess about the verbal stem for "to speak". Following the other examples we have, I could see 'igil, 'ugul, or 'egel. Another possibility is 'agal, but that seems less likely as a verbal form since we probably have 'agl, 'agal, or 'âgal meaning "word, utterance".
I looked extensively to see if there was any way that the -ish- was perhaps some kind of suffix for 'igl. I thought it might be a sort of pronominal suffix used to form the construct to pair it with mêk, which would then mean either "gesture" or perhaps "hand". Another possibility is that it is a suffix that marks the gerund (verbal noun). Also, I considered that 'igil was "speech" and shimêk was "gestures" or "hands", and that some kind of metathesis was occurring as a result of placing the words into a compound. However, I have found absolutely no basis for any of these possibilities in Hebrew or Arabic. As such, my best guess is that ishmêk is "gestures", following the references in The War of the Jewels. Arabic has plurals that have a pattern of 'VCCVC, where the root is prefixed with a glottal stop, and we can see such words with uzbad and inbar from Khuzdul. They occur in Adunaic as well.
My opinion is that 'igl is a gerund (verbal noun) meaning "speaking, speech" and is singular, nominative, composition.
Ishmêk is then probably "gestures" and plural, nominative, indefinite.
The War of the Jewels, pg 395, 402