My Shelley had something similar - granted she was a sprayed female Irish Setter dog, I'm not too sure how this would translate to cats? But we ended up using an OTC salt water nasal spray (obviously NOT recommended for cats as they're too small) twice a day at morning and night for about a week, and it seemed to help! For cats, maybe you could look for something similar? I don't honestly know if there's anything out there like that. Or you could try a humidifier (doesn't have to be a big one) with sea salt (NOT table salt - if you crush up sea salt into a super fine powder some of it does go into the vapor that's put out - ever been by the beach and you can smell the salt in the air?) and a drop or two of peppermint oil? Maybe put her in the bathroom with it on for a 1/2 hour at a time and see? I don't know. We did that with Annie, our Aussie Shepard.
Both helped to unclog the nose, and gave relief to the lungs. Not a cure, but they helped ease the symptoms. Since cats are smaller I wouldn't suggest the humidifier method for longer than a 1/2 hour at a time, since it'll add more moisture to the lungs. So long as she DOESN'T have any forms of pneumonia it should be ok. Also if you DO use oils to humidify be VERY careful as some can be really harmful, especially if they're ingested or if too much is used to put into the air. Depending on the size of your bathroom and the size humidifier, I'd suggest 2 drops peppermint oil, 1 drop eucalyptus and if you want to soothe or calm her, 1 drop lemongrass and/or 1 drop lavender. Sea salt should be about 1 tablespoon (crushed to powder) per liter or 1.5 liters.
Mix well with a spoon before adding it to the humidifier - it helps to warm it a little in the microwave or stovetop first.
I do know that prolonged coughing CAN make them throw up - the diaphram is so active and working so hard it can tighten the muscles in the core and abdominal region, making them throw up even though it's unconnected to an infection, etc. and since it sounds like your baby was coughing a lot for a good long time, that's what I'd suspect. Then again, if she WAS coughing things up but swallowing them, given enough volume that could make her throw up too potentially.
Do you know if her coughing was dry, or wet? Especially the sound of it - if it's dry (sounds kinda of like a "gheeeeee" or a bit rattly then she probably has phlegm/mucus that's trying to bust up and come out (which is where the infection will be trapped so added moisture in the air helps) but if it sounds wet (noisier almost like a suction-y gurgley sound) then that's a bit more worrying to me. If she can't get that out it's a good breeding ground for whatever she has the longer it stays in, so added moisture might NOT help.
As for drinking and feeding, you might need to assist feed - or maybe try some tricks? For my Shelley and Annie they didn't want to eat either because they couldn't smell it but they DID know what their food looked like so I always let them see me scooping out the kibble or decanning their wet food. Then I'd scoop a tiny bit on my finger and open their mouth, rubbing it on their tongue. They couldn't taste too well but they knew the texture. It didn't always work, but it did seem to help. Another trick, fat. If you cook up a chicken, turkey or any part there of, the fat that's rendered - smear some on your finger and rub that on their gums/tongue, they might know the feel and it might perk them up.
What's the weather been like too in your area? Dry, humid, heavy pollen, pollution, etc? Those can have some effects too while she's sick. Windows open, closed? A/C or furnace on? Try to limit cleaning products with noticeable smells if you can, limit any chemicals period even if your place is well ventilated. What this will do, is limit the exposure to more inflammatory causes that might slow down or impede the meds she's on. You don't want any further irritation if you can help it. No perfumes, colognes, air fresheners, etc. This goes for laundry detergent too. For Shelley we bought a box of DREFT - it's a newborn/baby/kid type detergent that seemed to work ok.
I agree on the worry of her not eating - that would be concern #1, and for quicker results I'd think the IV fluids would be best but also the more expensive choice as opposed to the subQ.
Another question - if your vet gave her IV fluids, did they notice any change in her peeing? For my Shelley when she got IV fluids, her peeing slowed WAY down because her body soaked it all up so well to work with it as opposed to drinking from a bowl. She never got subQ so unsure how she would've reacted. After 2 days of straight IV fluids, she started peeing a little more often, and a little more volume each time as her body finally had it's fill and started cycling it as it should. Her vet at the time (now passed away) called it the "suck up" phase - he said, "there's dehydration and then there's DEHYDRATION" and sometimes you never know until IV fluids are given and they're monitored. IV fluids immediately get circulated, subQ takes awhile to absorb so it can be harder to tell.
PS - I DON'T like Baytril, while it's a good antibiotic it can be harsh on the body at that kind of dose if you're giving it for the first time. If it nor the Covenia work, ask for something similar and see if you can maybe break up the needed dose into multiple doses a day, that might help her appetite too. For my Shelley, I just looked up her records, she was getting 250mg of it (she was 101 lbs) and it effected her plus it started to wreck some havoc with her bowels - depending on how long this goes on (the infection) maybe ask about a probiotic.
Again this was with my dogs, but might be things to talk to your vet about. Love to you, your family and your baby!!
PPS - if you DO use nasal drops or sprays: administer it from BEHIND them. Tuck them up against you, their back to your front. Using your non-dominate hand, cup them under their chin and tilt their head up - not a full "to the sky" but a good upward angle. Just as you're about to give the meds in 1 nostril, using a finger of your non-dominate hand, cover their free nostril when they breathe out just as you give the dose. This'll make them want to INHALE instead of EXHALE so they'll get more of the medicine. Keep their head up for a few seconds and they WILL squirm around, then let their head go but keep holding them, talk to them, pet them, etc. Then do the other side if that's what the meds say.
Also with nasal sprays and drops they WILL get a runny nose for a bit afterwards, this is normal and should stop within about 10-15 min. The discharge SHOULD be clear - if it's yellow looking or darker, any odd shade of greenish-brown (up to reddish looking) it could indicate a sinus infection as well.
They WILL hack/gag a bit - this is also normal, you're not suffocating them if giving the right dose, as directed, it's just their body reacting to something unfamiliar but if it gets worse, contact your vet.
**EDIT - just found something else in my dogs records, for Annie when she was stuffed/plugged up (along with URI) it was inflammation plus mucus/discharge once the treatments started working so the vet gave us a "newborn infant" like bulb syringe so we could suction out her nostrils when needed. If that becomes the case, you can definitely internet search!
Thanks for sharing.