Standoff at Sea
After failing to defeat the British fleet on January 26th, the Comte de Grasse (at right) kept the British navy hemmed in along the southeastern coast of St. Kitts. Each day, the French ships came within sight of the British fleet, and more it looked as if a major attack might commence. The only real combat that occurred, however, was an occasional clash involving a few frigates, schooners, or other, smaller vessels. 
De Grasse was frustrated with this business. His fleet had no proper anchorage, and the constant patrols at sea wore down his ships and crew. His vessels ran out of their original store of provisions in early February, and the crews then subsisted on provisions seized or commandeered from merchant ships. The French fleet was also low on ammunition after the several battles with the British on January 25th-26th. De Grasse seems to have feared being caught in this situation once an expected British reinforcement (Admiral George Rodney’s squadron) arrived and made the British fleet larger than his own. 
Around the time that the original provisions gave out, de Grasse was arguing that the further prosecution of the siege was inadvisable. However, the Marquis de Bouillé, who commanded the French troops on land, was determined to continue. Through some mysteries grapevine, the British naval officers almost immediately learned of this division and it gave them fresh hope. On February 8th, Captain Robert Manners of the Resolution wrote:
I understand the French commanding officers are all at variance. De Grasse is not for risking his squadron, probably wishing to preserve it for the more important conquest of Jamaica. The Marquis de Bouille declares he will not give the island up, though Dr Grasse should leave him, and [Comte de] Bougainville sides with De Bouille… 
De Grasse relented and maintained a thankless watch on the British fleet.
Battered Brimstone Hill
The Marquis de Bouillé’s determination to maintain the siege was well founded. On land the French were at last making good progress battering Brimstone Hill into submission.
On the night of January 31st, French infantry found at the base of Brimstone Hill a large, abandoned cache of artillery. This included eight brass 24-pounders, with 6,000 cannonballs, and two brass 13-inch mortars, with 1,500 shells. These guns had been part of the “travelling artillery of the West Indies,” kept in storage on St. Kitts. As the artillery belonged neither to the British garrison, nor to the island’s militia, neither had thought to remove the guns and mortars to a more secure location when the French invasion began. This windfall allowed the ammunition-starved French army to escalate its bombardment of the British garrison. 
A few days later (February 3), de Bouillé was reinforced with a battalion of Regiment Hainault, dispatched from Grenada. Also, the ship of the line Caton lent two of its 18-pounders and ten of its 24-pounders to the besieging army. 
The Marquis de Bouillé (at center, holding sword) at the siege of Brimstone Hill. In the background are burnt-out houses in the town of Sandy Point. At right, a mortar battery fires on the hill. (Excerpt of a French illustration).
Artillery are dragged forward during the siege of Brimstone Hill. (Excerpt of a French illustration).
The British naval commander, Vice Admiral Samuel Hood, could do little to aid the garrison. Brigadier-General Robert Prescott had returned to Antigua after the inconclusive battle on January 28th. Hood wrote that on February 8th, he was informed by signals from Brimstone Hill “that the enemy’s batteries had been successful in damaging the works and buildings [on the hill], [and] that the garrison was reduced and short of ordnance stores.” Therefore, he dispatched several officers to provide moral support to the garrison, but although the men went at night and worse disguises, all were captured. 
By February 11th, the state of the garrison had grown quite grim. Governor Thomas Shirley noted in his journal:
[The French] opened a battery of 4 guns near… the foot of the Hill, against the north-west front, from whence they very much annoyed the garrison on the highest parts. Twenty-three pieces of cannon and all their mortars were this day incessantly played upon the Hill, whereby the breaches already made were greatly widened and the garrison became much reduced by killed and wounded. 
Matters were even worse on the 12th. Shirley wrote:
This day, on the northwest front was an entire breach and all the guns disabled. In the curtain were two very large breaches; the whole parapet was destroyed… In the left flank all the guns were disabled and in the left face was a practicable breach of forty feet. 
Lieutenant George Lewis Hamilton described the woeful condition of the garrison’s artillery:
Upon our opening the batteries on the lower works, on the first appearance of the enemy, there was two twenty four pounders, four twelve pounders, two nine pounders and one eight inch howitzer mounted… The progress of the enemy, since they have opened their gun batteries, has been so heavy and rapid… that the eight inch howitzer only remains serviceable, and from the present ruinous and exposed state of the whole front[, it] can only be brought into action in the night, when it is supposed that the enemy's fire has abated and they are making approach to assault. 
He found the situation to be little better in the upper citadel and concluded that the “guns and carriages… are in the worst state and are absolutely insufficient to prevent the approaches of the enemy.”
The French could clearly see the breeches in the fortress walls, and the Marquis de Bouillé decided to mount an assault. He wrote:
The day of the 13th was to be used to reconnoiter and make dispositions, and the attack was to be on the 14th, one hour before daylight. The Marquis de Chilleau, one of the bravest men that I have known, was to command the head of my attack. The grenadiers and soldiers were full of ardor, and although I assumed that I would lose many, I counted on success. 
1. John Gilmary Shea (1864). The operations of the French fleet under the Count de Grasse in 1781-2 as described in two contemporary journals; John Ross (1838). Memoirs and correspondence of Admiral Lord de Saumarez, Vol. 1.
2. Shea, ibid.
4. The Journal Politique for 1782; Attaque et prise de Saint-Christophe dit le «Gibraltar» des Antilles (janvier-février 1782), in Revue Historique des Armées, Vol. 1, 1974. This article includes extracts from «Mémoires Secrets» de Bouillé; The Remembrancer, Vol. 14.
5. Attaque et prise..., ibid.
6. Attaque et prise..., ibid. Charles Middleton (1907). Letters and papers of Charles, Lord Barham, Volume 1. David Hannay (1895). Letters written by Sir Samuel Hood (Viscount Hood) in 1781-2-3.
7. Algernon Aspinall (1915). West Indian tales of old.
8. Attaque et prise..., ibid. (Translation is my own)