It should be clear that medicine administered through a nebuliser, enters the lungs and stomach via the nose and the mouth. Thus, using a nebuliser while fasting invalidates the fast.
As for the situation described, if the woman is fully incapable of fasting, is sick to the extent that fasting could further worsen her health and has no hope of regaining her health and the ability to fast, she can compensate for the missed fasts by giving fidyah. The amount she must give for fidyah is the same as the amount given for sadqa al fitr, which is 1.75 kg of wheat or money equivalent to the price of 1.75 kg of wheat.
It is important to keep in mind that if the woman somehow regains her health and the ability to fast later on in her lifetime, it will become obligatory for her to perform qaza for the missed fasts, despite the fact that she has already given fidyah.
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دلائل:
القرآن الکریم: (البقرة، الآیۃ: 184)
فَمَنْ کَانَ مِنْکُمْ مَّرِیْضاً اَوْ عَلٰی سَفَرٍ فَعِدَّةٌ مِّنْ اَیَّامٍ اُخَرَ وَعَلَی الَّذِیْنَ یُطِیْقُوْنَهٗ فِدْیَةٌ طَعَامُ مِسْکِیْنٍ....الخ
الدر المختار: (422/2، ط: سعید)
فصل فی العوارض المبیحہ لعدم الصوم۔۔۔۔۔۔۔او مریض خاف الزیادۃ لمرضہ وصحیح خاف المرض۔۔۔۔۔او مریض خاف الزیادۃ او ابطاء البرء او فساد عضو۔۔۔۔۔
رد المحتار: (395/2، ط: سعید)
قولہ (أنہ لو أدخل حلقہ الدخان) أی بأی صورۃ کان الإدخال، حتی لوتبخر ببخور فآواہ إلی نفسہ واشتمہ ذاکراً لصومہ أفطر۔لإمکان التحرز عنہ وھذا مما یغفل عنہ کثیر من الناس، ولا یتوھم أنہ کشم الورد ومائہ والمسک لوضوح الفرق بین ھواء تطیب بریح المسک و شبہہ وبین جوھر دخان وصل إلی جوفہ بفعلہ۔
Allah the Almighty knows best.
Darul ifta Al-Ikhlas, Karachi.