To summarize the discussion that has happened in the above comments:

1. There is no general single-word replacement for "part of speech" that has any real usefulness.
2. If you want the average English speaker to understand what you mean, just say "part of speech".
3. If you are speaking with academic grammarians, it may be more appropriate to use the term "word class", and in some very specific circumstances, if everyone involved is paying attention, you may be able to get away with just "class", which is a one word replacement, though it is really just an abbreviated form of the longer, two-word term.

Short Answer

Since you don't like it, you might say one of the following, depending on how opinionated you want to be (more opinionated can be considered more impolite or more "honest", depending on how you view things and who you are talking to).

I don't like the [description of smell] of the hospital:

• smell, strong smell
• chemical smell, antiseptic smell
• nauseating smell, sickeningly sweet smell

Objective Descriptions

A personal affinity for a smell is often subjective, as demonstrated in this dialog:

"I don't like the smell of the hospital."

"What smell?

"The unpleasant smell."

Stating that one "doesn't like it" means that it's "unpleasant", so that doesn't provide any new information. You may want an objective word to describe the smell other than the unpleasantness. But the more accurate you are, especially for uncommon smells, the less likely it might be understood by the listener.

"I don't like the iodoform smell of the hospital."

"Huh?"

"Iodoform is the disinfectant they use here that gives this hospital that distinctive smell. I don't like that.

This above effect may be desirable. If not, you can use a more generic description:

"I don't like the disinfectant smell of the hospital."

"Oh that's a disinfectant? I didn't know that. It's kind of strange, but it doesn't bother me."

Or you can go even more generic, as follows:

"I don't like the (strange, strong) chemical smell of the hospital."

"Yes, I've noticed that too. It's kind of strange, but it doesn't bother me." Thnking: Hmmm I never thought about it being a chemical. I suppose it is...

And the most generic is simply to indicate the smell as "the one that is currently observable in the environment":

"I don't like the smell of this hospital."

"I don't smell anything." OR "It doesn't bother me." OR "Neither do I."

More elaborate options for literary effect.

One way to gain more insight and options in language-use is to perform a Google search using a "wildcard" asterisk, which will match anything, as demonstrated in Google Search: "the*smell of the hospital":^1,2

• antiseptic (10 results)
• sterile (4 results)
• strong (3 results)
• Other results: overpowering, poisonous, putrid, clean, peculiar, unmistakable, crisp, pungent, sour, unique, fresh.

The above sample, from the first few result pages, indicates "antiseptic" and "sterile" as common descriptions.³ The smell is likely caused by various disinfectants, most notably iodoform (though the use of that chemical is declining in favor of more odorless disinfectants).

The words you use to describe this depends on the goal of your speech or writing, keeping your audience in mind. Not every hospital has the same smell, and people react differently to it - some even enjoy the smell. The most broadly acceptable description would be "hospital smell" since that allows each person to "fill in the blank" with their own experience, and so its the most accurate description from an audience's experiential point of view. However, if you're writing a crime drama, you might use more colorful words to engage the reader's senses and heighten corresponding emotions and themes:

[Original text written for this answer.]

As Jenny crawled in the blackness of the dirty, underground, rat-maze prison, she came upon a silhouette of light behind a cold metal door. Once upon a time, such a light would be a reassuring welcome that no monsters were around. But here it was exactly the opposite: the light meant the monster was real.

From the edges of the cold, steel door, a familiar smell overcame Jenny with an unfamiliar shock. The clean, reassuring hospital smell she knew as a nurse was completely out of place here. She suddenly realized why her patients didn't like it. To them, it was not a smell of daily routine, it was a smell of sickness and mortality. This was not a place one wanted to be. The strong, antiseptic smell reeked of the unpleasant possibilities of this place. The monster was in the smell, in the light, behind that door.

^{1. Internet search engine results can be helpful to provide quick answers to language usage. But these results can also be highly inaccurate, and cannot be relied upon to provide statistical significance of actual usage. For more accurate analysis of language usage, one needs to refer to various corpora (link).}

^{2. It's interesting to note that one gets slightly different Google results from "the*smell of the hospital" and "the * smell of the hospital". The latter has spaces around the wildcard.}

^{3. Keep in mind note 1.}