How are you managing to keep the sick harlequin's jug of water at the appropriate temperature? Wrapping bubble wrap or a small towel or facecloth round the outside of the jug, but making sure it doesn't topple over, might help a little bit to conserve the water at a particular temperature - but this certainly isn't ideal.
As you've no filter in there, then it's probably best to do a water change of this jug water daily, perhaps scooping out some of the water with an old yoghurt pot or cup, then replacing the removed water with dechlorinated water of the required temperature. Dose the medication after you've done the water change. It's probably much too risky to try to replenish medication lost through the water changes (i.e. trying to judge how much has been discarded and adding that to the new day's dosage), so just dose whatever amount is appropriate for the particular day and the jug water volume.
As for a hospital/quarantine tank, your sister's original tank (the 19-litre one which she was going to upgrade from) would make an ideal hospital/quarantine tank and give the harlequin a better opportunity than a jug - is that a possibility for you? You could then use the spare filter of your mum's friend for now, or an alternative to buying an additional filter is an airstone for the quarantine tank - again not ideal but definitely preferable to having no quarantine/hospital set-up at all. That just leaves a heater to get - could you take on some extra odd jobs to finance that?