My home temple's bell broke by falling on the ground while cleaning it. Does it mean any inauspicious thing is going to happen ? Is it inauspicious to repair it and use it again ?
No it is not inauspicious. Repair and use it. I have given below an incident in Sri Ramakrishna's life where he repaired an image of Krishna. The repaired image is still worshipped. You can still see the image of Krishna if you go to Dakshineswar temple.
One day the priest of the Radhakanta temple accidentally dropped the image of Krishna on the floor, breaking one of its legs. The pundits advised the Rani to install a new image, since the worship of an image with a broken limb was against the scriptural injunctions. But the Rani was fond of the image, and she asked Sri Ramakrishna's opinion. In an abstracted mood, he said: "This solution is ridiculous. If a son-in-law of the Rani broke his leg, would she discard him and put another in his place? Wouldn't she rather arrange for his treatment? Why should she not do the same thing in this case too? Let the image be repaired and worshipped as before." It was a simple, straightforward solution and was accepted by the Rani. Sri Ramakrishna himself mended the break. The priest was dismissed for his carelessness, and at Mathur Babu's earnest request Sri Ramakrishna accepted the office of priest in the Radhakanta temple.
The Gospel of Sri Ramakrishna, Introduction, Kali by Swami Nikhilananda
Till now I have written about what should be done. I have not explained why it is a correct thing to do.
Image worship is based on the principle of love.
Master (to the teacher): “What is wrong with image worship? The Vedanta says that Brahman manifests Itself where there is ‘Existence, Light, and Love’. Therefore nothing exists but Brahman.”
The Gospel of Sri Ramakrishna, With the Devotees at Dakshineswar (II)
The reason for scriptural injunction against broken image is to prevent disrespect to the deity. Sri Ramakrishna could look into the heart of a person and saw that Rani Rashmoni was not guided by any disrespect but love. Hence He repaired the broken image.
The main point is that it is never good to mechanically follow scriptural injunction. It is better to understand the reason for the injunction. The spirit of the injunction is more important than the letter of the injunction.
In the present case of a broken bell the questioner is not showing any disrespect. Hence it is perfectly all right to repair and use the bell.