You seem to view this as a specific characteristic of the English language. I'm not familiar with your native language in particular, so I may be wrong, but I expect you have this view precisely because English is not your native language. I don't know of a language where the words are spoken in complete isolation - spaced out by pauses to the extent that the word boundaries could easily be spotted on a spectrogram, and phonetically completely unaffected by the words uttered before and after. In general, we hear pauses between words that aren't really there, because we recognize the words we're already familiar with, whereas natural speech in a completely novel language will almost always sound like an endless stream of linked words.
Thus, the distiction that is worth making, the way I see it, is not between how to teach English as opposed to another given language, or how to teach a second as opposed to a first language, but rather, what to think about when teaching a child a language by immersing them in the language, as opposed to a formal education setting. Any advice I have to give is from that angle, and my personal experience of it has been limited to teaching a first language (and it wasn't English), so it mainly pertains to younger ages, although as the language is novel to the child, I can see that it would still apply:
- Speaking slowly and clearly is generally viewed as helpful. You may take that as an indication to "not link".
- With newborns acquiring their first language, we generally speak in overly simplified sentences. Not "look at her eyes", but "Look. Eyes." This may be another indication to not link, but I don't really see how it transfers to older children. If you were practicing One parent One language from the beginning, I would say that you should both talk like that in your respective languages. Suddenly switching to such simplified speech with a 2 year old may be confusing. But then again, certainly not more confusing than suddenly switching to a language the child doesn't understand.
- You can assist the child in finding the word boundaries in your speech by modulating tone and volume to emphasize key parts of the message: /həraaaɪz/. With the very young, this is especially helpful as variable pitch, elevated pitch overall, and elongated vowels (the manner of speech colloquially called "parenteese") helps maintain the child's interest in the communicative effort.
- You can further help the child understand which part of the message is associated with which emphasized phrase by utilizing of baby signing, for example, pointing to your eye during the /aɪz/ part of the utterance.
So much on what you can do. You should also note the following, regarding natural language acquisition by mere exposure:
- Learning is not linear. From your concern, you seem to hold out hopes that you will be able to give your child a slowly accumulating but always accurate vocabulary. This is not a likely trajectory. Mistakes are normal and entirely to be expected. In formal education, you get to say "these are the 20 vocables we'll be practicing today", and getting to master those in isolation. With natural acquisition, you're practicing everything all the time, with slowly accumulating prowess.
- The example you mention seems a bit contrived to me, but for instance, expect a period where overgeneralization is super common (e.g. "runned" instead of "ran", even though "runned" is a word form the child has never been exposed to). Don't be alarmed by language misuse, even if your child should indeed present with constructions such as /hɪz-raɪz/.
- Model correct pronounciation. Don't make the child feel that language is difficult or not for them. Don't say "No, it's not /raɪz/ it's /aɪz/", you say "Yes, I see /hɪz aaɪɪz/ too." Just keep exposing the child to the correct pronounciation, and they will pick up.